From the Residue Table to Divisor Sums: Exact Row Identities in the Number Periodic Table

This manuscript is the inaugural paper in a series dedicated to the foundations of the Number Periodic Table (NPT) research program. It isolates the exact local law governing the structural transitions within the NPT, shifting the mathematical focus from asymptotic estimates to deterministic row identities. For a given row i ≥ 0 and column j ≥ 1, the framework defines two primary states: the ordinary residue R(i,j) = i j, and the positive distance to the next multiple of j, defined as $M(i,j) = j - R(i,j)$. Every column follows a deterministic countdown and reset cycle, starting from a synchronized base state at row $i=0$ where $R(0,j) = 0$ and $M(0,j) = j$. For i, j ≥ 1, the central result is the exact adjacent-row identity: I(i,j) = 1 + M(i,j) - M(i-1,j) = j · 1j i By removing the universal background countdown of $-1$ from every column, the remaining corrected impulse exactly identifies divisibility. Key Results & Findings Divisor Recovery: The single row-transition identity directly recovers the divisor set of i, the divisor count τ(i), and the divisor sum σ(i) without relying on statistical models. Local and Cumulative Identities: By defining a row sum M(i) = ∑j ≤ i M(i,j), the paper establishes the exact local identity σ(i) = M(i) - M(i-1) + i - 1. Cumulative Ledger: The framework provides an exact cumulative formula for the sum of all previous divisor sums: ∑m ≤ i σ(m) = M(i) + i(i-1)/2. Structural Primality Criterion: The absence of an interior column reset defines primality; for i ≥ 2, i is prime if and only if M(i) - M(i-1) = 2. Finite Validation: The theoretical claims are separated from conjectural bounds and independently verified by an exact finite audit. The audit checks all 2,001,000 corrected impulses through row $2000$, within a total of 2,015,000 exact checks, with zero failures. Formal Certification: A companion Lean 4 formalization machine-checks the corrected impulse identity and its finite divisor-sum and divisor-count corollaries. It imports only Std, requires no Mathlib, and contains no sorry, admit, or user-declared axioms. As the first installment in the NPT foundational series, this paper serves as the foundational text for subsequent topics, providing a precise, unified local law for use in broader investigations of number behaviorism and divisor problems.