Simultaneous amplification beyond fitness three halves

Alec Kriebel

doi:10.5281/zenodo.21852072

Abstract

A single graph family, chosen without knowing mutant fitness, can simultaneously amplify fixation under Birth–death and death–Birth updating beyond fitness \(3/2\). The construction combines a large clique core, ordinary hub leaves, and dilute internally heavy vertex pairs. Exact trace equations and uniform error estimates show that every fixed \(1<r<R_{\mathrm{hyb}}\) is eventually amplified under both rules.

New lower bound\(R_{\mathrm{sim}}\ge 1.5028569127905696\ldots>3/2\)

One familyGraph sizes and weights depend on the population index, never on fitness

Global statusThe proposed \(3/2\) threshold is false; the exact unrestricted value remains open

Main theorem

Threshold-breaking family. Let

\[ P(R)=R^6-8R^5+22R^4-30R^3+21R^2-6R+1, \]

and let \(R_{\mathrm{hyb}}=1.5028569127905696267\ldots\) be its unique root in \((3/2,151/100)\). There is an explicit fitness-independent family \(G_t\) such that, for every fixed \(1<r<R_{\mathrm{hyb}}\), both normalized fixation probabilities are strictly greater than one for all sufficiently large \(t\).

The order of quantifiers is essential: the same graph sequence works for the whole open fitness interval, while the population threshold may depend on the fixed fitness. This proves \(R_{\mathrm{sim}}\ge R_{\mathrm{hyb}}\) and therefore exactly refutes the candidate equality \(R_{\mathrm{sim}}=3/2\).

The dilute pair–leaf construction

At index \(t\), take a clique core of size \(C_t=t^4\), put \(m_t=\lfloor\lambda_*t\rfloor\) unit-weight leaves on one core hub, and add \(q_t=t\) disjoint two-vertex satellites. Each satellite edge has weight \(C_t/\sigma_*\); every satellite vertex has the same positive weak edge to every core vertex; distinct satellites have no edges between them. The weak edge is selected by an effective least-dyadic compact-uniform diagonal, so every finite graph is connected while the separated trace remains asymptotically exact.

The fixed algebraic parameters are

\[ \sigma_*={-R^3+4R^2-3R-1\over2(R-1)}, \qquad \lambda_*={2(1-\sigma_*)(R-1)\over1+\sigma_*(R^2-1)} \]

evaluated at \(R=R_{\mathrm{hyb}}\). Their numerical values are \(\sigma_*=0.130677282287048\ldots\) and \(\lambda_*=0.750806483031880\ldots\).

For each fixed fitness, the normalized Bd and dB fixation probabilities have the exact first-order forms

\[ \begin{aligned} x_t&=1+{q_t\over C_t}\left[ {2(\sigma_*-1)\over1+\sigma_*(r^2-1)} +{\lambda_*\over r-1}\right]+o(q_t/C_t),\\ y_t&=1+{q_t\over C_t}\left[ {2(r(2-r)-\sigma_*)\over\sigma_*+2r(r-1)} -\lambda_*\right]+o(q_t/C_t). \end{aligned} \]

Exact monotonicity and tangency certificates prove that both bracketed coefficients are positive precisely throughout the claimed common interval for these fixed parameters.

Exact verification

Labelled-chain audit: all 512 labelled states of a finite audit graph are aggregated into 108 orbit fibres, and every Bd and dB transition agrees exactly with the five-coordinate lumped formulas.
Coefficient replay: independent symbolic derivations reproduce both pair-gate odds, the leaf corrections, the rational endpoint margins, and the optimized coefficients.
Algebraic threshold: a Sturm certificate isolates the unique sextic root, while exact derivative and discriminant identities prove tangency and interval monotonicity.
Full fixation path: logarithmic core cutoffs, gambler's-ruin products, cleanup blocks, and reciprocal-invasion bounds control establishment through fixation at the strict \(o(q_t/C_t)\) scale.
Independent asymptotics: a second least-integer diagonal reaches the same trace and threshold through a separately organized limiting argument.
Clean replay: the release archive contains pinned dependencies and a bootstrap that reproduces every finite exact certificate from a clean extraction.
Website integrity: the served PDF and social-preview image are pinned by SHA-256.

Finite transition checks validate the implementations. The growing-family quantifiers are discharged by the analytic trace and uniform error estimates, not by enumeration or sampled fitness values.

A global affine route is also closed

An independent exact order-\(22\) weighted graph has normalized endpoint values \(x=0.9334417185\ldots\) and \(y=1.0336117074\ldots\), with \((x+2y)/3>1\). Its exact crossing coefficient is \(0.3355466820\ldots>1/3\). Separate clique–pendant asymptotics force any universal convex affine separator to use a Bd coefficient at most \(1/3\). Consequently no graph-independent convex affine endpoint separator of the form \(\theta x+(1-\theta)y\le1\) exists.

This finite affine witness is not itself a simultaneous endpoint amplifier; the dilute pair–leaf family supplies the actual threshold-breaking simultaneous construction.

Scope and open problems

The exact value \(R_{\mathrm{hyb}}\) is class-optimal for the displayed dilute \(K_2\)-satellite plus ordinary-leaf leading regime.
Uniform-portal unweighted gadgets through six vertices, dilute clique satellites, and several weighted-leaf scalings have exact obstruction certificates and do not improve this mechanism.
No matching universal upper bound is known. In particular, the unrestricted exact value of \(R_{\mathrm{sim}}\) remains open.
A separate fitness-two program has an exact marked-chain and sum-of-squares reduction, but its final stationary promotion inequality is not proved.

AI-assisted research disclosure

OpenAI language models were used extensively in exploration, proof development, software generation, verification design, literature organization, and drafting. Alec Kriebel directed the work, selected the released claims, and is responsible for the decision to publish these materials. Model output was not treated as evidence by itself.

Suggested citation

Alec Kriebel, “Simultaneous amplification beyond fitness three halves,” version 1.0.0, 8 August 2026. doi:10.5281/zenodo.21852072.

Contact

Alec Kriebel

Independent Researcher

Technical correspondence: me@aleckriebel.com

ORCID 0009-0001-9320-500X