The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent
Abstract
We study the density ρ(N,c) of distinct entries of the N×N multiplication table in the square-gap ((cN)², (cN+1)²], equivalently the density of integers near (cN)² possessing a divisor within a factor 1/c of their square root. Exact computation is carried to N = 10^8, an exact-integer Monte Carlo to N = 2^1024, and a continuum model of factorizations, anchored to the exact tiers at seven seam heights, to N = 2^{10^18}. All predictions were frozen before their data and every failure is logged. Three results emerge. First, square-centered windows are statistically indistinguishable from generic windows of the same magnitude, and the gap-to-gap coefficient of variation fits N^β with β = -0.440 ± 0.031, so the unproven localization of Ford's theorem to √x-scale windows is empirically a concentration statement with a square-root exponent. Second, the local decay of ρ attains Ford's shape -δ - (3/2)t^{-1} in t = log log x only from t ≈ 13–16, and the exponent δ = 0.086071 itself is never attained. At t = 39.5, the model height of quintillion-bit integers, the effective exponent still reads 0.077. The deficit follows a 1/t law with coefficient b = 0.333 ± 0.032 in the mature regime, a constant floor being rejected at Δχ² = 56.8. Third, an exactly computable ballot skeleton, Ford's order-statistics mechanism stripped of all divisor structure, reproduces δ, the t^{-3/2}, and the 1/t coefficient to within 1–2 standard errors. The lossy steps in the reduction from divisors to the ballot event therefore carry nothing at subleading order, and the divisor-specific contribution is localized to a transient dying near t ≈ 25. The subleading law is a measurement in the model, anchored to exact integers at the seams. It is not a theorem, and we state the theorem-shaped objects it calls for.
Related
- The Profile Calculus: An Axiomatic Certification Framework for the ω ≤ 9/4 Matrix-Multiplication Exponent
- Entropy Concentration and Universal Typicality for Almost i.i.d. Quantum Sources
- Closed energy formulas on tensor squares of perfect crystals of types D4(3) and G2(1)
- Latin squares as a model of perfect mixing of high-entropy alloys
- Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields
BibTeX
@misc{ross2026multiplication,
title = {{The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent}},
author = {Michael Ross},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23483059},
url = {https://zenodo.org/records/23483059},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent
AU - Michael Ross
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23483059
UR - https://zenodo.org/records/23483059
AB - We study the density ρ(N,c) of distinct entries of the N×N multiplication table in the square-gap ((cN)², (cN+1)²], equivalently the density of integers near (cN)² possessing a divisor within a factor 1/c of their square root. Exact computation is carried to N = 10^8, an exact-integer Monte Carlo to N = 2^1024, and a continuum model of factorizations, anchored to the exact tiers at seven seam heights, to N = 2^{10^18}. All predictions were frozen before their data and every failure is logged. Three results emerge. First, square-centered windows are statistically indistinguishable from generic windows of the same magnitude, and the gap-to-gap coefficient of variation fits N^β with β = -0.440 ± 0.031, so the unproven localization of Ford's theorem to √x-scale windows is empirically a concentration statement with a square-root exponent. Second, the local decay of ρ attains Ford's shape -δ - (3/2)t^{-1} in t = log log x only from t ≈ 13–16, and the exponent δ = 0.086071 itself is never attained. At t = 39.5, the model height of quintillion-bit integers, the effective exponent still reads 0.077. The deficit follows a 1/t law with coefficient b = 0.333 ± 0.032 in the mature regime, a constant floor being rejected at Δχ² = 56.8. Third, an exactly computable ballot skeleton, Ford's order-statistics mechanism stripped of all divisor structure, reproduces δ, the t^{-3/2}, and the 1/t coefficient to within 1–2 standard errors. The lossy steps in the reduction from divisors to the ballot event therefore carry nothing at subleading order, and the divisor-specific contribution is localized to a transient dying near t ≈ 25. The subleading law is a measurement in the model, anchored to exact integers at the seams. It is not a theorem, and we state the theorem-shaped objects it calls for.
ER -