Topological Collapse and Asymmetric Divergence of Carmichael Pseudoprimes on the 6N Skeleton

On the 6N skeleton every integer coprime to 6 sits on a wing: the right wing 6N+1 (≡1 mod 6) or the left wing 6N−1 (≡5 mod 6). Reading the Carmichael numbers through this geometry, Korselt's criterion forces three elementary but rigid structural laws: (i) a one-way semipermeable membrane — every left-wing Carmichael number has all of its prime factors on the left wing, and is not divisible by 3; (ii) a parity law — a Carmichael number coprime to 6 lies on the left wing iff it has an odd number of left-wing prime factors; and (iii) a unified barrier — whenever 3 ∤ (n−1), no right-wing prime can divide n, fusing the left wing with the off-skeleton residue-3 numbers (a residue-3 Carmichael number is exactly a 3 glued to a pure-left core). Enumerating all 1547 Carmichael numbers up to 1e10, all three laws hold with zero violations. The right-to-left ratio R(X) climbs monotonically from 5.0 at X=1e4 to 88.8 at X=1e10, and the same monochromatic suppression acts inside the right wing, where the pure-right family is overtaken by the mixed family — the internal ratio M(X) crossing unity between 1e7 and 1e8. We are explicit about the limits: the two single-colour basins decay in the same direction but at materially different rates (M ~ (log X)^−1.1 vs R ~ (log X)^+2.9, a ~2.6× gap traced to the smoothness of p−1 for right primes versus its roughness for left primes), so no single thermodynamic law governs them, and neither asymptotic rate is identifiable under Poisson sparsity and the deep pre-asymptotic character of Carmichael statistics. What is unified is algebraic — the single barrier 3 ∤ (n−1) — not thermodynamic. Includes the full reproducer (with three-law verifiers and the M(X) split), data (CSV), figure, and paper.

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Zenodo (CERN European Organization for Nuclear Research)
Published
2026-06-09
DOI
https://doi.org/10.5281/zenodo.20611583
Primary Topic
Quasicrystal Structures and Properties
Type
preprint
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Topological Collapse and Asymmetric Divergence of Carmichael Pseudoprimes on the 6N Skeleton

Ruqing Chen
Zenodo (CERN European Organization for Nuclear Research)
Quasicrystal Structures and Properties
preprint

Topological Collapse and Asymmetric Divergence of Carmichael Pseudoprimes on the 6N Skeleton

Ruqing Chen
preprint en

Abstract

On the 6N skeleton every integer coprime to 6 sits on a wing: the right wing 6N+1 (≡1 mod 6) or the left wing 6N−1 (≡5 mod 6). Reading the Carmichael numbers through this geometry, Korselt's criterion forces three elementary but rigid structural laws: (i) a one-way semipermeable membrane — every left-wing Carmichael number has all of its prime factors on the left wing, and is not divisible by 3; (ii) a parity law — a Carmichael number coprime to 6 lies on the left wing iff it has an odd number of left-wing prime factors; and (iii) a unified barrier — whenever 3 ∤ (n−1), no right-wing prime can divide n, fusing the left wing with the off-skeleton residue-3 numbers (a residue-3 Carmichael number is exactly a 3 glued to a pure-left core). Enumerating all 1547 Carmichael numbers up to 1e10, all three laws hold with zero violations. The right-to-left ratio R(X) climbs monotonically from 5.0 at X=1e4 to 88.8 at X=1e10, and the same monochromatic suppression acts inside the right wing, where the pure-right family is overtaken by the mixed family — the internal ratio M(X) crossing unity between 1e7 and 1e8. We are explicit about the limits: the two single-colour basins decay in the same direction but at materially different rates (M ~ (log X)^−1.1 vs R ~ (log X)^+2.9, a ~2.6× gap traced to the smoothness of p−1 for right primes versus its roughness for left primes), so no single thermodynamic law governs them, and neither asymptotic rate is identifiable under Poisson sparsity and the deep pre-asymptotic character of Carmichael statistics. What is unified is algebraic — the single barrier 3 ∤ (n−1) — not thermodynamic. Includes the full reproducer (with three-law verifiers and the M(X) split), data (CSV), figure, and paper.

Zenodo (CERN European Organization for Nuclear Research)
Energoservis (Czechia) (CZ)
Peace, Justice and strong institutions
Quasicrystal Structures and Properties
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