Review Papers Claim by Claim

1) What the paper does (claims and methods)

• The authors derive modified Friedmann equations in Horndeski (scalar-tensor) and in bi-scalar-tensor constructions, define effective dark-energy density and pressure and an effective Newton constant G_eff, then numerically integrate the cosmological evolution chosen so that CMB-era H(z_CMB)≈H_LCDM while allowing deviations at low redshift to raise H0 (Friedmann structure and G_eff
• Two mechanisms are highlighted: (A) decreased G_eff at intermediate redshifts produced by chosen G5(X) friction terms in Horndeski models, (B) phantom effective dark-energy equation-of-state (w_DE<-1) in bi-scalar-tensor models; explicit models and parameter ranges are given and numerically shown to reach H0≈74 km/s/Mpc for particular parameters (Summary of mechanisms

2) Main results (quantitative)

• Horndeski example: with G5(X)=ξX^2 and K=-V0 φ+X the authors numerically find that H0 can reach ≈74 km/s/Mpc for ξ in a range (quoted 1.2–1.7 in H0 units) while matching ΛCDM at z_CMB (Horndeski numeric example
• Bi-scalar examples: Model I (K=1/2 φ - ζ^2 B, G=0) and Model II (K=1/2 φ, G=ξ B) produce phantom w_DE for parameter choices like ζ≈-10 or ξ≈-10 (Planck units) and yield H0≈74 km/s/Mpc due to phantom-driven faster late-time expansion (Bi-scalar numeric example

3) Strengths

• Clear theoretical mechanisms spelled out (G_eff reduction vs phantom w_DE) and explicit Lagrangians exhibiting them; numerical integrations shown in figures for representative parameter choices (Numerical demonstration
• Checks of linear perturbative stability (conditions c2_S≥0, Q_S>0) and discussion of GW speed c_T≈1 are included rather than ignored (Stability and GW speed

4) Weaknesses, blindspots, and critical caveats

1. Parameter tuning and naturalness: the parameter values quoted (e.g., ξ and ζ with extreme mass-dimension scalings in natural/Planck units) are not motivated by microphysics; the relevant dimensionful scales are enormously far from particle-physics scales, implying fine-tuning or lack of UV motivation (Dimensionful scale concern
2. Data confrontation absent: authors tune models to match H(z_CMB) and Ω_m0 and show H0 increases, but do not perform a full parameter estimation against Planck CMB, BAO, SNe, growth or lensing data. That raises risk of hidden tensions (e.g., sound-horizon constraints, BAO distances, growth σ8) that can invalidate nominal H0 improvement. The paper explicitly notes numerical examples rather than global fits (No global data fit
3. Perturbations and large-scale structure: while linear stability conditions are checked, there is no calculation of scalar perturbation transfer functions, matter power spectrum, or lensing potentials and their comparison with data; a reduced G_eff or phantom w_DE can alter growth history and may conflict with redshift-space distortion and weak lensing constraints (Growth and G_eff implications
4. Reproducibility: no public code or numerical data are provided; the paper reports numerical solutions but does not supply parameter chains or implementation details or data files, reducing reproducibility and independent validation (authors assert figures are from their numeric solutions) (Numerical solutions without public code
5. Fine tuning to satisfy cT≈1: following GW170817, viable modified gravity often requires structure that forces gravitational wave speed to equal c; authors note that this can be achieved but requires fine tuning in some cases and mention this repeatedly (GW speed fine tuning

5) How convincing is the result? (confidence and ways to falsify)

Reasonable theoretical plausibility but not yet observationally robust: the paper provides consistent theoretical models and numerical examples (moderate internal consistency), but lacks global data fitting and perturbation-level observational tests — so the H0 alleviation claim is suggestive not decisive.

Falsification tests that would disprove the models

• If full CMB+BAO+SNe+RSD parameter estimation excludes the parameter windows that produce H0≈74 at >3σ, models are falsified in practice.
• If detailed growth and lensing predictions for the models conflict with observed fσ8 and lensing power (Planck, DES/KSZ, KiDS) the mechanism is observationally excluded.
• If future gravitational-wave speed limits tighten further around c with uncertainties below the model-induced deviations and the required fine-tuning becomes implausible, the model class will be strongly constrained.

6) Concrete, practical suggestions to advance this work

1. Perform full Monte Carlo parameter estimation (Planck high-ell + low-ell, BAO, Pantheon SNe, SH0ES R22/R23, RSD/fσ8) with the models implemented in a Boltzmann solver (e.g., hi_class or EFTCAMB) and publish chains and code. That will test whether H0 increase survives other probes.
2. Compute matter power spectrum, CMB lensing, ISW and compare to data — particularly check the scale dependence induced by G_eff(k,z) (the paper provides G_eff formula but not full transfer computations) (G_eff expression
3. Publish the numerical implementation and input files used to reproduce Figures 1–4 (initial conditions, units and conversion details for Planck vs H0 units) to enable reproducibility.

7) Short checklist for readers

8) Final, balanced conclusion

The paper provides useful theoretical constructions and concrete numerical demonstrations that modified-gravity late-time mechanisms (reduced G_eff or phantom effective dark energy) can in principle raise H0 to values compatible with local measurements while maintaining linear stability and potentially c_T≈1. However, because the work stops short of confronting the full suite of cosmological data and of releasing code, the result is an important theoretical contribution and a well-motivated invitation to further observational tests, not a definitive solution to the H0 tension. Key next steps are full Boltzmann-level implementation, global MCMC fits, and public release of the numerical code and chains.

9) Buttons and actionable next steps