A mathematical analysis of slow Fibonacci walks has found a sharp split in a conjecture about their gap patterns: the down-integer and up-integer sets have exactly the same third-order gap spectrum, but they part company at the fourth order. At that level, the up-integer set contains the gap value 9 while the down-integer set does not.
The result is based on deterministic proofs that track coefficients through maps Ls and show those maps converge uniformly on T to an invertible limiting map. The mathematical domain is every integer n≥2 with a unique canonical representation n=aF_t+bF_{t−1}, with t≥2 and 1≤a≤b≤F_t.
A conjecture holds, then fails
In this setting, a gap spectrum is the collection of gap values that occur at a specified order of the walk. The third-order result is an exact match: both D3 and U3 contain 3, 4, 5, 6, 7, 8, 10, 11 and 13, with no other values in the stated spectrum.
The fourth-order picture is different. The shared values are 5, 6, 7, 8, 10, 11, 13, 16 and 18, but U4 has one additional value, 9. The refined fourth-order class makes the distinction especially concrete: D4(9) is empty, whereas U4(9) contains the anchor 8.
Together, the two findings test the proposed identity Dℓ=Uℓ for all ℓ. The analysis supports that identity at order 3 and disproves it at order 4, so agreement at one level cannot be extended across all orders on the basis of these results.
The counts follow a repeating phase pattern
The paper next turns from which gaps occur to how often refined occurrences appear. For every fixed pair of indices ℓ,m≥1, the normalized counts are governed by a continuous phase profile, Pℓ,m. The profile repeats when its argument is multiplied by ϕ^4, while the down- and up-integer counts use phase arguments that differ by a factor of ϕ^2.
That phase profile gives the same set of possible subsequential limits for the two sequences: Pℓ,m([1,ϕ^4]). In other words, the normalized counts can trace a recurring range as the scale changes, even though the two constructions reach the profile at different phase arguments.
The framework distinguishes ordinary natural density from logarithmic density. The two refined sets have matching lower and upper densities; natural densities exist exactly when the relevant phase profile is constant, while logarithmic densities always exist and agree.
For the general case, the common logarithmic density is specified by an integral of gℓ,m(θ) divided by θ(1−θ), taken from α to β and scaled by 1/(4 logϕ). The formula provides the analytic link between the phase profile and the density of a refined class.
First-order classes show the same tension
At first order, the analysis finds that only gap values belonging to D1 produce nonempty refined classes. For those values, the corresponding down- and up-integer classes have positive lower density, do not have a natural density, and share the same logarithmic density. First-order values outside D1 produce empty refined classes in both sequences.
The study reports exact logarithmic-density expressions for the first-order gap values 1, 2, 3 and 5. These are the explicitly evaluated first-order cases, while the general treatment is expressed through phase-profile functions and integrals.
The first-order result shows that a refined class can have positive lower density without its ordinary proportion settling to a natural density. In these cases, the analysis instead supplies a common logarithmic density for the down- and up-integer classes.
A result within one mathematical system
The conclusions are tied to canonical Fibonacci representations and the derived down- and up-integer sets. They do not establish Dℓ=Uℓ for every order: the fourth-order result already supplies a counterexample through the gap value 9. Nor do they show that all refined sets have natural densities; that occurs only when the relevant phase profile is constant.
The supplied version is arXiv version 1 dated 20 Aug 2026, and the authors declare no conflict of competing interest.
Paper data and sources
Original title: Gap spectra and densities of slow Fibonacci walks
Authors: Yaping Mao, Qinghong Zhao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text