The Reduced (⌊6k/5⌋+1)/5 Collatz-Type Domain with Complete 5-Adic Reduction and One Observed Attractor up to 10^11

This record presents a detailed mathematical and computational study of the reduced Collatz-type map generated by

F(n) = floor(6n/5) + 1,

followed by complete removal of every factor of 5. The reduced transformation is therefore

T(n) = F(n) / 5^ν₅(F(n)),

where ν₅(m) denotes the 5-adic valuation of m.

The map contains the exact cycle

1 → 2 → 3 → 4 → 1,

which is the only attractor observed in the exhaustive computation reported here.

A complete first-descent verification was performed for every positive integer

1 ≤ n ≤ 10^11.

For every tested starting value n ≥ 5, an iterate strictly smaller than n was reached. The maximum first-descent stopping time was

τ_max = 422,

first attained at

n = 55,335,461,598.

The maximum elementary stopping count was 470. The largest observed 5-adic valuation was

ν₅ = 16,

and the largest intermediate state encountered was

8,669,800,813,122,806,007,604,

arising from the starting value

n = 37,266,730,278.

The computation processed 100,000,000,000 starting values in 4,000 independent chunks. The mean first-descent stopping time was

6.18073627485.

The observed fraction of steps at which 5-adic reduction was activated was

0.1999996001463434,

extremely close to 1/5.

The mean number of divisions by 5 per growth step was

0.24999981555069667,

extremely close to 1/4.

The accompanying article develops the algebraic, modular, 5-adic, inverse-branch, cycle, and probabilistic structure of the map. It derives exact residue conditions for reduction, studies hypothetical periodic orbits through Diophantine identities, analyzes inverse branches, and develops a probabilistic model for the 5-adic reduction process.

A strict distinction is maintained between exact algebraic results, exhaustive finite-range computation, and probabilistic interpretation.

The reproducibility package contains the complete 4,000-chunk computational certificate set, SHA-256 manifests, summary statistics, stopping-time histograms, record tables, production C++ source code, aggregation and verification tools, execution logs, and the LaTeX source of the article.

The exhaustive computation establishes that every positive integer n ≤ 10^11 eventually enters the cycle

1 → 2 → 3 → 4 → 1.

Equivalently, convergence to 1 is computationally verified throughout the entire tested range.

The broader global convergence conjecture states that every positive integer eventually enters the same cycle. This conjecture remains open beyond the verified range.