Negative ResultPaper P06

Why Uniform k CANNOT Resolve H0/S8 Tensions

An honest negative result: Non-Newtonian Calculus with uniform k values does NOT fix cosmological tensions.

Scientific Honesty: Publishing negative results prevents wasted effort and shows what approaches do NOT work.

Why Negative Results Matter

Science advances not just by discovering what works, but by rigorously documenting what does NOT work. Negative results prevent others from wasting time on dead ends and demonstrate scientific integrity.

What This Shows

  • - We tested the hypothesis rigorously
  • - We report results honestly, even when negative
  • - We save others time by showing this path fails
  • - We maintain scientific credibility

What Would Be Wrong

  • - Hiding failures and only showing successes
  • - Cherry-picking favorable results
  • - Overclaiming based on weak evidence
  • - Ignoring contradictory findings

Background: Cosmological Tensions

1

H0 Tension (Hubble Constant)

5.7-sigma discrepancy in expansion rate measurements

Early Universe (CMB - Planck)

H0 = 67.4 +/- 0.5 km/s/Mpc

Late Universe (Local - SH0ES)

H0 = 73.04 +/- 1.04 km/s/Mpc

This 5.6 km/s/Mpc difference (8%) suggests either new physics or systematic errors.

NASA: Webb & Hubble Confirm Expansion Rate Puzzle
2

S8 Tension (Matter Clustering)

2-3 sigma discrepancy in matter distribution

CMB Prediction (Planck)

S8 = 0.834 +/- 0.016

Weak Lensing Observed (KiDS)

S8 = 0.759 +/- 0.024

The universe appears less clumpy than predicted by early universe measurements.

eROSITA and S8 Tension (2024)

The Hypothesis We Tested

Hypothesis:

Modified Friedmann equations using Non-Newtonian Calculus with a single uniform k value can simultaneously resolve both H0 and S8 tensions by providing scale-dependent corrections.

Result: REJECTED

After systematic testing with multi-objective optimization (9 Pareto solutions tested), we found that uniform k values CANNOT resolve these tensions. The framework produces uniform corrections that shift both tensions in the same direction, failing to bridge the gap.

Why Uniform k Cannot Work

The Problem

H0 and S8 tensions require OPPOSITE corrections:

  • - H0 needs early values to INCREASE or late values to DECREASE
  • - S8 needs early values to DECREASE or late values to INCREASE
  • - A uniform k applies the same scaling everywhere
  • - This creates a uniform shift, not differential correction

What Would Be Needed

To resolve tensions, you would need:

  • - Scale-dependent k(z) varying with redshift
  • - Different k values for different physics (H0 vs S8)
  • - A physical mechanism for why k should vary
  • - Independent evidence for the mechanism

Our framework does NOT provide this - uniform k is too simple.

The Evidence: MOO Results

We used multi-objective optimization (pymoo NSGA-II) to search for k values that could minimize both chi-squared (fit to data) and fragility (numerical stability). The results are clear:

0/9

H0 Resolutions

0/9

S8 Resolutions

-28%

Li7 Change

Interactive Evidence

The visualizer above shows the Pareto front from our multi-objective optimization. Notice how NO solution successfully bridges both H0 and S8 gaps simultaneously - uniform k simply cannot provide the differential corrections needed.

What This Teaches Us

Lesson 1: Test Your Hypotheses

We could have assumed NNC would work for cosmology based on other successes. Instead, we tested it rigorously and found it fails. This is how science should work.

Lesson 2: Report All Results

Publishing this negative result is as important as publishing positive results. It shows the boundaries of where NNC applies and prevents publication bias.

Lesson 3: Understand Limitations

The framework works well for problems with uniform scaling (crack tips, radiation transport) but fails for problems requiring differential corrections (cosmological tensions). Knowing this helps us apply it appropriately.

Lesson 4: Honest Communication

We clearly state what does NOT work, not just what does. This builds trust and credibility in our positive results by showing we are willing to admit failures.

NNC: Where It Works vs Where It Fails

Where NNC Works

  • Crack tip singularities: 93.4% improvement with k=-0.5
  • Radiation transport: 83.6% improvement with k=-1.0
  • Vortex cores: Uniform scaling with k=-1.0
  • Power-law singularities: Where D*[1/r] = -1 helps

Common pattern: Problems with localized singularities requiring uniform regularization

Where NNC Fails

  • H0/S8 tensions: Need differential corrections
  • Quantum mechanics: k=0 (classical) optimal
  • Smooth diffusion: No singularities to regularize
  • Problems requiring varying k: Single k too rigid

Common pattern: Problems requiring scale-dependent or differential corrections

Future Directions

While uniform k fails, this negative result suggests productive directions:

  1. 1. Investigate k(z) profiles:

    Can redshift-dependent k values resolve tensions? What physical mechanism would justify this?

  2. 2. Focus on localized applications:

    Use NNC where it works (singularities, fracture) rather than forcing it on cosmology.

  3. 3. Develop theoretical foundation:

    Derive k(scale) from first principles rather than fitting to data.

  4. 4. Document boundaries:

    Systematically map where NNC helps vs hurts to guide appropriate application.

Related Pages

Cosmological Tensions

Existence proofs for parameter fits (not resolutions) - what we CAN claim

CASCADE Singularity Results

Where NNC DOES work: 93.4% improvement on crack tip singularities

Lessons Learned

12 years of failures and what they taught us about honest science

What We CAN Verify

Spectral gap preservation and constancy diagnostic - rigorous positive results

References

[Planck 2020] Planck Collaboration, Aghanim, N., et al. (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6.

[Riess et al. 2022] Riess, A. G., et al. (2022). A Comprehensive Measurement of the Local Value of the Hubble Constant. The Astrophysical Journal Letters, 934, L7.

[KiDS 2021] Heymans, C., et al. (2021). KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints. A&A, 646, A140.

[Grossman & Katz 1972] Grossman, M. & Katz, R. (1972). Non-Newtonian Calculus. Lee Press, Pigeon Cove, MA.

[Blank & Deb 2020] Blank, J. & Deb, K. (2020). Pymoo: Multi-Objective Optimization in Python. IEEE Access, 8, 89497-89509.