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arxiv: 2603.03113 · v3 · submitted 2026-03-03 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Emergent ΛCDM cosmology from a measure-induced deformation of the Newtonian action

S. M. M. Rasouli
Authors on Pith no claims yet

Pith reviewed 2026-05-15 16:45 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Newtonian actionfractional kerneleffective cosmological constantFLRW backgroundmeasure deformationLambdaCDM emergencealpha deformation
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The pith

A minimal time-dependent deformation of the Newtonian action produces the full matter, radiation, and accelerated phases of LambdaCDM cosmology from one potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces the ordinary time integral in the Newtonian action with a fractional kernel controlled by a single parameter alpha. This change induces both a friction-like term in the equations of motion and an alpha-dependent effective gravitational potential. The resulting background equations match the Friedmann-Lemaître-Robertson-Walker evolution, so that radiation domination and late-time acceleration appear automatically once alpha departs slightly from its Newtonian value of 1. Standard Newtonian cosmology cannot generate those two epochs, yet here they arise together with matter domination without adding separate energy components. The deviation |alpha minus 1| must remain small to match observations, and this same deviation sets the size of the emergent cosmological constant.

Core claim

Replacing the standard time integration measure in the Newtonian action by a time-dependent fractional kernel with parameter alpha yields a conserved quantity that includes a memory-like kinetic term. Generalizing the Newtonian potential to an alpha-dependent effective potential then produces cosmological equations whose background dynamics are identical to those of the relativistic FLRW metric. With a single unified potential the model self-consistently reproduces the matter-dominated, radiation-dominated, and present accelerated epochs, while an effective cosmological constant emerges whose magnitude is set by the small departure of alpha from 1. When alpha equals 1 the construction re-rec

What carries the argument

The alpha-dependent effective potential induced by the fractional time kernel, which replaces the Newtonian potential and supplies the effective cosmological constant when alpha differs slightly from 1.

Load-bearing premise

The alpha-dependent effective potential obtained from the measure deformation produces Friedmann-like background equations that correctly describe the full expansion history.

What would settle it

A direct integration of the derived equations showing that no value of alpha near 1 can simultaneously fit supernova distances, CMB peak positions, and the matter-radiation equality redshift would falsify the claimed correspondence.

read the original abstract

We propose a minimal extension of the Newtonian action by introducing a time-dependent fractional kernel characterized by a single deformation parameter $\alpha$. This kernel admits a natural interpretation as a nontrivial temporal integration measure defined by a time-dependent kernel, placing the formulation within measure-based approaches to anomalous or fractal dynamics. Despite the appearance of a friction-like term in the equations of motion, a conserved quantity is still obtained, containing a memory-like fractional kinetic energy contribution. Moreover, by generalizing the standard Newtonian potential to an $\alpha$-dependent effective potential induced by the underlying measure, the resulting cosmological equations exhibit an effective correspondence with relativistic FLRW cosmology at the level of background dynamics. In the limit $\alpha=1$, the framework reduces to standard Newtonian cosmology. We show that, with a single unified potential, the matter-dominated, radiation-dominated, and present accelerated phases are obtained self-consistently, while the latter two epochs cannot be described within standard Newtonian cosmology. The structural presence of $\alpha$ in all physical observables allows theoretical and observational constraints to be imposed, indicating that compatibility with observational data requires $|\alpha - 1|\ll1$. Within this framework, an effective cosmological constant emerges, controlled by the small deviation of $\alpha$ from the Newtonian limit. These results demonstrate that $\Lambda$CDM cosmological dynamics emerge from a simple measure-induced deformation of the Newtonian action.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a minimal extension of the Newtonian action by introducing a time-dependent fractional kernel with a single deformation parameter α, interpreted as a nontrivial temporal integration measure. Generalizing the standard Newtonian potential to an α-dependent effective potential induced by this measure yields cosmological equations that correspond to FLRW background dynamics, including an emergent cosmological constant proportional to (1-α). With a single unified potential, the framework self-consistently reproduces matter-dominated, radiation-dominated, and accelerated phases, reducing to standard Newtonian cosmology at α=1, with observational compatibility requiring |α-1|≪1.

Significance. If the mapping from the fractional kernel to the effective potential is shown to be rigorously derived rather than imposed, the result would be significant for demonstrating how ΛCDM dynamics, including the accelerated phase inaccessible in standard Newtonian cosmology, can emerge from a measure-based deformation of Newtonian mechanics. The single-parameter unification of epochs and the structural role of α in observables provide a falsifiable framework with clear observational constraints.

major comments (2)
  1. [Generalization of the Newtonian potential and background equations] The central claim of FLRW correspondence rests on generalizing the Newtonian potential to an α-dependent effective potential 'induced by the underlying measure.' The explicit term-by-term derivation from the time-dependent fractional kernel to this potential form is not provided in sufficient detail; without it, the potential appears selected to enforce the target Friedmann and acceleration equations (including Λ ∝ (1-α)), rendering the emergence circular rather than independently derived. This is load-bearing for the self-consistent reproduction of all epochs with one potential.
  2. [Derivation of conserved quantity and cosmological equations] The equations of motion include a friction-like term yet yield a conserved quantity with memory-like fractional kinetic energy. The manuscript must demonstrate explicitly how this conserved quantity, combined with the deformed potential, produces the exact FLRW set without additional ansätze, particularly for the radiation and accelerated phases. The reduction at α=1 is automatic, but the necessity of the (1-α) deviation for Λ must be shown from the kernel rather than by construction.
minor comments (1)
  1. Clarify the notation for the fractional kernel and its time dependence early in the text to aid readability for readers unfamiliar with measure-based anomalous dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the derivations require more explicit detail to avoid any appearance of circularity and will revise the manuscript accordingly. We address each major comment below.

read point-by-point responses

  1. Referee: [Generalization of the Newtonian potential and background equations] The central claim of FLRW correspondence rests on generalizing the Newtonian potential to an α-dependent effective potential 'induced by the underlying measure.' The explicit term-by-term derivation from the time-dependent fractional kernel to this potential form is not provided in sufficient detail; without it, the potential appears selected to enforce the target Friedmann and acceleration equations (including Λ ∝ (1-α)), rendering the emergence circular rather than independently derived. This is load-bearing for the self-consistent reproduction of all epochs with one potential.

    Authors: We acknowledge that the term-by-term mapping from the fractional kernel to the α-dependent effective potential was not presented with sufficient intermediate steps. In the revised manuscript we will insert a new subsection deriving the effective potential explicitly from the time-dependent kernel, showing each contribution that generates the (1-α) correction. This derivation establishes that the potential form is fixed by the measure deformation rather than chosen to match the target equations. With this addition the single-potential reproduction of all epochs follows directly from the equations of motion. revision: yes

  2. Referee: [Derivation of conserved quantity and cosmological equations] The equations of motion include a friction-like term yet yield a conserved quantity with memory-like fractional kinetic energy. The manuscript must demonstrate explicitly how this conserved quantity, combined with the deformed potential, produces the exact FLRW set without additional ansätze, particularly for the radiation and accelerated phases. The reduction at α=1 is automatic, but the necessity of the (1-α) deviation for Λ must be shown from the kernel rather than by construction.

    Authors: We agree that the passage from the conserved quantity to the full set of FLRW equations needs to be expanded. The revised version will include a complete, step-by-step derivation that begins from the memory-like fractional kinetic term, incorporates the deformed potential, and arrives at the background equations for matter, radiation, and accelerated epochs without introducing extra assumptions. The (1-α) term for the effective cosmological constant will be traced directly to the kernel structure, with the α=1 reduction shown as the limiting case of that same derivation. revision: yes

Circularity Check

2 steps flagged

α-dependent effective potential and emergent Λ are introduced to enforce FLRW match by construction

specific steps
  1. self definitional [Abstract]
    "by generalizing the standard Newtonian potential to an α-dependent effective potential induced by the underlying measure, the resulting cosmological equations exhibit an effective correspondence with relativistic FLRW cosmology at the level of background dynamics. ... an effective cosmological constant emerges, controlled by the small deviation of α from the Newtonian limit."

    The effective potential is presented as induced by the measure, yet its α-dependence is defined precisely so that insertion into the action produces the Friedmann and acceleration equations with an emergent Λ term proportional to (1-α). The 'emergence' therefore reduces to a reparametrization of the input deformation rather than a derived consequence.

  2. fitted input called prediction [Abstract]
    "We show that, with a single unified potential, the matter-dominated, radiation-dominated, and present accelerated phases are obtained self-consistently, while the latter two epochs cannot be described within standard Newtonian cosmology."

    The unified potential is chosen to reproduce all three epochs simultaneously; the reproduction is therefore a consistency check on the imposed functional form rather than a prediction independent of the choice of potential.

full rationale

The derivation introduces a time-dependent fractional kernel with parameter α, then posits an α-dependent effective potential 'induced by the underlying measure' whose explicit form is selected so that the deformed Newtonian action yields background equations identical to FLRW (including a term proportional to (1-α) acting as Λ). The reduction of the equations to the target cosmology is therefore enforced by the choice of generalization rather than obtained as an independent output of the kernel; the single unified potential is likewise tuned to recover matter, radiation, and accelerated phases simultaneously. This matches the self-definitional pattern where the deformation parameter directly parametrizes the desired emergent feature.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The model relies on one free parameter α and the assumption that the measure induces a specific effective potential; no additional entities like new particles are introduced.

free parameters (1)
  • α = |α - 1| << 1
    Single deformation parameter of the kernel, constrained by compatibility with observational data to |α - 1| << 1.
axioms (1)
  • domain assumption The time-dependent fractional kernel can be interpreted as a nontrivial temporal integration measure.
    This places the formulation within measure-based approaches to anomalous dynamics.
invented entities (1)
  • α-dependent effective potential no independent evidence
    purpose: To produce the effective correspondence with relativistic FLRW cosmology at background level.
    Generalized from the standard Newtonian potential induced by the measure.

pith-pipeline@v0.9.0 · 5543 in / 1447 out tokens · 46952 ms · 2026-05-15T16:45:12.104650+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    by generalizing the standard Newtonian potential to an α-dependent effective potential induced by the underlying measure... an effective cosmological constant emerges, controlled by the small deviation of α from the Newtonian limit

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We consider a new interpolate fractional potential as Φ_α(a;α)=R(α)a^{-2}W_rad(a)+L(α)a^2 W_Λ(a)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Evolution of density perturbations in fractional cosmology

    gr-qc 2026-03 unverdicted novelty 6.0

    Fractional cosmology modifies density perturbation growth via parameter α, yielding a new upper bound α ≲ 1.07 from σ8 normalization and Sachs-Wolfe constraints.

  2. Emergent inflation in fractional cosmology

    gr-qc 2026-03 unverdicted novelty 6.0

    Fractional cosmology produces emergent inflation as a stable attractor from a non-singular pre-inflationary regime, with the number of e-folds related to the fractional parameter α and a subsequent radiation-dominated era.

Reference graph

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