A 5^r-Scaled Family of the Reduced (5k-1(3))/4 Collatz-Type Map with Two Observed Attractors

This work develops a 5^r-scaled family of the reduced 5k-1(3)/4 Collatz-type map previously studied on the positive odd integers. For each integer r >= 0, the state space is restricted to the scaled odd domain S_r = 5^r N_odd, and the residue-dependent affine correction is scaled by the same factor 5^r. The two branches are therefore 5x - 5^r for x congruent to 1 modulo 4 and 5x - 3*5^r for x congruent to 3 modulo 4, followed in each case by division by the complete power of two contained in the affine numerator.

The central result is an exact algebraic conjugacy between the base system and every member of the scaled family. If T_0 denotes the base reduced map and T_r the scaled map, then

T_r(5^r y) = 5^r T_0(y),

and, more generally,

T_r^j(5^r y) = 5^r T_0^j(y)

for every nonnegative iterate j.

This conjugacy shows that the scaled parameter r does not generate independent dynamics on the scaled domain. The two observed base fixed points 1 and 3 are transported exactly to 5^r and 3*5^r. Periodic orbits correspond bijectively under multiplication by 5^r, with minimal periods preserved. Branch sequences, two-adic valuation sequences, capture times, first-descent stopping times, basin membership, peak iteration indices, and normalized orbit profiles are invariant under the conjugacy, while absolute state values, first-lower values, reduced peaks, and raw affine numerators are multiplied by 5^r.

The paper also derives the scaled periodic-orbit identity, the corresponding necessary mean-valuation restriction for positive cycles, and the exact scaled inverse branches and inverse forests. Numerical examples for several values of r are included as direct illustrations of the algebraic scaling identities.

The exhaustive finite verification from the base reduced 5k-1(3)/4 study tested all 50,000,000,000 positive odd starting values below 10^11. Every tested orbit entered one of the two observed fixed points 1 or 3, with zero additional terminal cycles and zero unsigned 128-bit overflow events. The exact basin counts were 32,318,909,987 for 1 and 17,681,090,013 for 3, corresponding to 64.637819974% and 35.362180026%, respectively.

These finite results are not presented here as a new 10^11 computational scan. Instead, the exact conjugacy transfers the complete verified base result to the corresponding scaled set consisting of the numbers 5^r y with y positive, odd, and below 10^11. Thus, for every fixed r, the corresponding set contains exactly 50,000,000,000 scaled starting states and inherits the same finite basin counts and orbit-statistic structure.

The global convergence question remains open. The statement that the two displayed fixed points are the only attractors for all positive states in the relevant domain is not claimed as a theorem. For every fixed r, the global two-attractor conjecture for the scaled system is exactly equivalent, through algebraic conjugacy, to the unresolved conjecture for the base system.

The accompanying computational and reproducibility archive contains the LaTeX source, an independent scaling-consistency verifier, a machine-readable statement of the scaling rules and inherited base statistics, documentation, and SHA-256 checksums. The archive does not represent a new exhaustive scan; it provides reproducible checks of the exact scaling identities established in the paper.

Related base work: Zenodo record 22752810.