Generated Arithmetic and Histories

Classical natural-number constructions determine values and their arithmetic. This construction also retains how a value was produced. The distinction is mathematical: two expressions can agree in value while carrying different internal observables and different responses under assembly.

From correlation to arithmetic

Minimal unbiased compositional change selects a rotation plane and a Gaussian law. Local information and normalized total correlation determine the finite window used in the construction. Preservation of constants selects the Neumann closure within the separated homogeneous local boundary class. Its spectral responses order an initially unlabelled family of modes.

After orientation, pointwise modulation and composition define the two native operations. Modulation by the first positive mode gives successor. These operations yield an ordered Peano semiring with unique prime factorization. Winding degree then supplies its numerical certification. The modes retain quadratic spectral energy and canonical scale and orientation transport.

What an arithmetic value leaves unresolved

Ordered expressions retain brackets, operation labels and ordered inputs. Every value has a finite expression fibre. A pure sum producing six has six unit leaves, while an expression multiplying three by two has five. Their root energy is the same, but the corresponding internal quadratic defects are thirty and thirty-one. These are distinct data on a common value fibre.

Equal root-channel weights and independent subhistory assembly determine the conditional history law used in the paper. Conditional expectation separates value observables from internal history differences. Exact variance and entropy identities quantify the information retained by this separation.

Histories affect subsequent response

The complete history space carries a nonnegative self-adjoint trace-class assembly operator. Its contraction evolution couples value coefficients to history differences. An exact matrix element at value four gives a history vector with zero initial value readout and nonzero readout after applying the assembly operator. Eliminating the history-difference sector produces an exact integral equation with memory.

This is the distinction from starting directly with the power maps of a circle. The circle gives the arithmetic value representation; the preceding source construction and the retained expression space supply response, internal variation and history-dependent evolution.

Read and verify

Keywords: natural numbers, generated arithmetic, arithmetic expressions, conditional expectation, trace-class operators, operator semigroups, memory, formalized mathematics.

Research subjects and results

Primary subjects

Construction of natural numbers; Natural-number objects; Peano arithmetic; Unique factorization; Arithmetic expressions; Spectral theory; Gaussian semigroups; Non-Markovian dynamics; Memory kernels; Conditional expectation; Trace-class operators; Formalized mathematics; 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

Universal algebra; Structural recursion; Spectral decomposition; Heat semigroups; Probability measures; Hilbert spaces; Orthogonal projections; Self-adjoint operators; Trace-class estimates; Contraction semigroups; Integral equations; Projection methods; Gaussian tail bounds; Lean theorem proving; Machine-checked proofs; 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

Spectral construction of natural-number arithmetic; Finite evaluation fibres of arithmetic expressions; Unique factorization on generated arithmetic modes; Exact separation of value and expression-history information; Nontrivial coupling between arithmetic values and expression histories; Trace-class operator from arithmetic expression assembly; Exact memory equation for arithmetic readout; Canonical scale and orientation transport of arithmetic; 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

自然数构造; 自然数对象; 皮亚诺算术; 唯一分解; 算术表达式树; 生成历史; 条件期望; 条件熵; 迹类算子; 非马尔可夫动力学; 记忆方程; 形式化数学; 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.