Alec Kriebel

A fitness-independent family of simultaneous amplifiers beyond relative fitness 3/2

By Alec Kriebel, independent researcher.
Reviewed edition published · Zenodo version 2.0.4; earlier named release 8 August 2026 · Unrefereed preprint

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Abstract

We study fixation from a uniformly random single mutant on finite connected loopless undirected weighted graphs under both Birth–death (Bd) and death–Birth (dB) Moran updating. Let R_sim be the supremum of R for which one fitness-independent graph family eventually amplifies fixation under both rules for every fixed r ∈ (1,R). Let R_hyb be the unique root in (3/2,151/100) of R⁶ − 8R⁵ + 22R⁴ − 30R³ + 21R² − 6R + 1. We construct one fitness-independent family that simultaneously amplifies every fixed 1 < r < R_hyb for all sufficiently large members of the sequence. Therefore R_sim ≥ R_hyb = 1.5028569127905696… > 3/2.

The construction combines a large clique with dilute hub pendants and dilute, internally heavy two-vertex satellites. An exact finite-state derivation of the weak-cut limiting trace, together with uniform establishment, cleanup, and reciprocal-invasion estimates at the scale of the eventual gain, reduces the two normalized fixation corrections to explicit response functions. Optimizing their common positivity interval yields the displayed sextic. The same analysis proves that R_hyb is optimal among fixed positive (σ,λ) in the displayed first-order dilute pair–pendant response model; a specialization with entirely rational edge weights already crosses 3/2. Thus 3/2 is not the exact simultaneous-amplification threshold. The unrestricted value of R_sim, and even a finite universal upper bound, remain open.

The idea in plain English

In a structured population, a new mutation can disappear or eventually replace every resident. The probability of taking over depends on which individuals can replace which others. It also depends on whether reproduction is chosen before a replacement site, or a vacancy is chosen before its neighbors compete to fill it.

This paper constructs one sequence of weighted population graphs that improves takeover probability under each of these two pure update rules. The entire sequence is chosen independently of the beneficial mutation’s relative fitness. For each fixed fitness strictly between 1 and about 1.5028569, all sufficiently large members improve on a well-mixed population of the same size. The required size can depend on the fixed fitness.

The graph combines a large well-mixed clique with two sparse kinds of peripheral structure. Heavy pairs favor death–Birth fixation, while leaves attached to one hub favor Birth–death fixation. Their small contributions can be balanced to make both improvements positive. Earlier work by Svoboda and colleagues established simultaneous amplification below relative fitness 1.2; a separate earlier construction by the author reached the open interval below 1.5.

The improvement above 1.5 changes what thresholds are possible, although the advantage becomes small as the population grows. This is a mathematical construction with highly separated edge weights. It supplies no empirical prediction or practical population-size requirement.

The precise result

Let R_hyb be the unique root in (3/2,151/100) of R⁶ − 8R⁵ + 22R⁴ − 30R³ + 21R² − 6R + 1. There exists one fitness-independent sequence of finite connected loopless undirected weighted graphs G_t such that, for every fixed 1 < r < R_hyb, there exists t₀(r) for which every t ≥ t₀(r) strictly amplifies uniformly initialized single-mutant fixation under each pure Bd and dB process. Each comparison uses the complete graph on the same n_t vertices and the matching update rule. Hence R_sim ≥ R_hyb = 1.5028569127905696… > 3/2.

Bd chooses a fitness-weighted parent and then an edge-weighted replacement target. dB first chooses a uniformly random death site and then a neighboring parent with probability proportional to fitness times edge weight. Their finite complete-graph baselines differ: ρ_Bd(K_n,r) = (1−r⁻¹)/(1−r⁻ⁿ), whereas ρ_dB(K_n,r) = ((n−1)/n)(1−r⁻¹)/(1−r⁻⁽ⁿ⁻¹⁾). Initialization is uniform over all vertices, with exactly one mutant and resident fitness one.

The sequence uses a unit clique of C_t = t⁴ vertices, m_t = floor(λ_* t) unit hub leaves, and q_t = t two-vertex satellites with internal weight C_t/σ_*. Every satellite vertex connects to every clique vertex by a common positive dyadic weight selected by an exact compact-interval decision procedure. All weights are fixed before r is quantified.

At scale q_t/C_t, the two normalized fixation corrections are B(r) = 2(σ−1)/(1+σ(r²−1)) + λ/(r−1) and D(r) = 2{r(2−r)−σ}/(σ+2r(r−1)) − λ. Their simultaneous positivity produces the sextic threshold. The fixed parameters giving this response-model tangency are approximately σ_* = 0.1306772822870484 and λ_* = 0.7508064830318805.

Within this displayed first-order dilute response model, R_hyb is optimal among fixed positive σ and λ. The rational choice σ = 19/137 and λ = 20/27 already yields an entirely rational-edge graph sequence with threshold (5069 + 12√147001)/6439 = 1.50176815223369… > 3/2.

Why the result holds

For fixed finite module sizes, an exact finite-state Schur complement resolves rare migration between the center and satellites. It uses local fixation probabilities to obtain a limiting macro chain while retaining possible reversals of a partially completed sweep.

The center-module argument controls initial establishment, later cleanup, and reciprocal invasion to errors smaller than the eventual gain q_t/C_t. An ordinary approximation with an unspecified error tending to zero would not suffice because the claimed gain itself tends to zero.

The two module types have opposite response signs under the two update rules. The proof balances their densities, optimizes the shared positivity interval, and then selects one positive weak-edge weight for each t on an expanding compact fitness interval. This supplies the required order: one family, then every fixed fitness, then a fitness-dependent size threshold.

Scope and limitations

The construction uses finite connected loopless undirected weighted graphs, uniform initialization of exactly one mutant, resident fitness one, a fixed mutant fitness r > 1, and the two specified pure Bd and dB rules. No guarantee is supplied for arbitrary initialization, directed or unweighted constructions, empirical evolutionary dynamics, or a process that randomly mixes the update rules.

The theorem concerns each fixed r in the open interval (1,R_hyb). Neutrality r = 1 and the upper endpoint r = R_hyb are excluded; no strict endpoint amplification is established. There is no size bound uniform near either endpoint and no guarantee for fitness varying with population size. The sequence need not contain a graph of every integer order.

For the unrestricted simultaneous-amplification threshold R_sim, the result is a lower bound. Exact optimality applies only to fixed positive σ and λ in the displayed first-order dilute pair–pendant response model. It does not establish optimality over all growing graph families, singular or size-dependent parameter choices, other modules, or higher-order boundary gains. This work supplies no universal upper bound.

For each fixed interior fitness, the normalized fixation gain has order Θ(t⁻³) = Θ(n_t⁻³/⁴) and tends to zero. Pair weights grow with the clique size, and the compact-interval construction may require very small positive weak edges. No useful finite-size onset, fixation-time bound, bounded weight ratio, or practical optimality is established.

The archived exact verifiers check finite labelled transition aggregation, symbolic response identities, rational margins, root isolation, and tangency. They do not machine-verify the manuscript’s weak-cut convergence or population asymptotics.

This is an unrefereed preprint developed with extensive generative-AI assistance. The author accepts responsibility for its claims and citations.

Status and AI use

Generative AI was used extensively in mathematical exploration, derivation, adversarial analysis, software development, manuscript preparation and subsequent review. AI review is not independent human or journal peer review. This is an unrefereed preprint without formal proof-assistant certification. The named author is responsible for the claims, materials and citations.

The reviewed edition is Zenodo version 2.0.4, published on 25 August 2026. An earlier named release of this same work was published on 8 August 2026 at DOI 10.5281/zenodo.21852072. These are two releases of one paper; the earlier date is not the publication date of version 2.0.4. The standalone manuscript is dated August 2026, and the copy retained inside its supporting archive is dated 22 August 2026.

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I welcome substantive collaboration on independent verification of the stochastic comparison and weak-cut arguments, assessment of the gain-scale error estimates, or development of a journal submission. Authorship would reflect actual research and writing contributions.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

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Alec Kriebel. (2026). A fitness-independent family of simultaneous amplifiers beyond relative fitness 3/2 (Zenodo version 2.0.4, published 25 August 2026) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22089807

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