Minimal Binary Multiple Windows on Odd Composite Survivors of Wheel 30

We introduce a binary-window framework for odd composite integers that survive wheel 30, that is, integers coprime to 2, 3, and 5. For such a composite integer n, we consider all contiguous binary windows in the binary expansion of n, reinterpret them as integers, and study the smallest window that is a multiple of a true factor of n. This defines a minimal multiple window Wmin(n) and induces a local/global dichotomy according to whether Wmin(n) < n or Wmin(n) = n. The wheel-30 restriction is used only as a domain reduction step. The central object is not a sieve in the Eratosthenes sense, but the arithmetic behavior of minimal binary windows on the reduced domain. Computationally, up to 10^6, every odd composite survivor of wheel 30 exhibited at least one factor-multiple window. Among the 188,170 composites in the tested range, exact visibility of at least one true factor occurred in 45.570495% of the cases. The dynamics split into a local regime, Wmin(n) < n, in 64.808418% of the cases, and a global regime, Wmin(n) = n, in 35.191582% of the cases. We also prove in Proposition 3 an exact prefix divisibility criterion for semiprimes. If n = pq with primes p < q, then for every k >= 1, p divides floor(n/2^k) if and only if p(q mod 2^k) < 2^k. This gives a complete characterization of prefix divisibility by the smallest prime factor. To study the smallest prime factor specifically, we introduce the refined observable Wmin^(p)(n), defined as the smallest binary window divisible by spf(n). In all tested cases up to 10^6, Wmin^(p)(n) existed for every composite survivor, and in all global semiprime cases the quotient Wmin^(p)(n)/spf(n) coincided exactly with the cofactor. In particular, among the 61,341 global semiprime cases found up to 10^6, this law held with empirical accuracy 100%.

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Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-04-14
DOI
https://doi.org/10.5281/zenodo.19571070
Primary Topic
semigroups and automata theory
Type
article
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article

Minimal Binary Multiple Windows on Odd Composite Survivors of Wheel 30

Ricardo Adonis Caraccioli Abrego
Zenodo (CERN European Organization for Nuclear Research)
semigroups and automata theory
article

Minimal Binary Multiple Windows on Odd Composite Survivors of Wheel 30

Ricardo Adonis Caraccioli Abrego
article en

Abstract

We introduce a binary-window framework for odd composite integers that survive wheel 30, that is, integers coprime to 2, 3, and 5. For such a composite integer n, we consider all contiguous binary windows in the binary expansion of n, reinterpret them as integers, and study the smallest window that is a multiple of a true factor of n. This defines a minimal multiple window Wmin(n) and induces a local/global dichotomy according to whether Wmin(n) < n or Wmin(n) = n. The wheel-30 restriction is used only as a domain reduction step. The central object is not a sieve in the Eratosthenes sense, but the arithmetic behavior of minimal binary windows on the reduced domain. Computationally, up to 10^6, every odd composite survivor of wheel 30 exhibited at least one factor-multiple window. Among the 188,170 composites in the tested range, exact visibility of at least one true factor occurred in 45.570495% of the cases. The dynamics split into a local regime, Wmin(n) < n, in 64.808418% of the cases, and a global regime, Wmin(n) = n, in 35.191582% of the cases. We also prove in Proposition 3 an exact prefix divisibility criterion for semiprimes. If n = pq with primes p < q, then for every k >= 1, p divides floor(n/2^k) if and only if p(q mod 2^k) < 2^k. This gives a complete characterization of prefix divisibility by the smallest prime factor. To study the smallest prime factor specifically, we introduce the refined observable Wmin^(p)(n), defined as the smallest binary window divisible by spf(n). In all tested cases up to 10^6, Wmin^(p)(n) existed for every composite survivor, and in all global semiprime cases the quotient Wmin^(p)(n)/spf(n) coincided exactly with the cofactor. In particular, among the 61,341 global semiprime cases found up to 10^6, this law held with empirical accuracy 100%.

Zenodo (CERN European Organization for Nuclear Research)
National Autonomous University of Honduras (HN)
Peace, Justice and strong institutions
Openalex Percentile: Top 11%
semigroups and automata theory
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