Radix-Agnostic Detection of Discrete Scale Invariance: A Certified Three-Stage Protocol and a Null Result from Planck 2018

In the paper's words

Discrete scale invariance (DSI) — invariance under a discrete set of rescalings x → λ^n x — predicts log-periodic oscillations whose angular frequency in the logarithmic coordinate u = ln x is ω₀ = 2π/lnλ. When the scaling radix λ is unknown, a radix-locked search at assumed radices risks both false negatives (the true radix was never probed) and multiplicity inflation (a continuum of probed frequencies is scanned without correction). This work develops and certifies a radix-agnostic detection protocol in three stages: (1) detrend in log-space and locate the spectral peak by FFT or Lomb–Scargle periodogram; (2) refine the peak with a bounded sinusoid fit on the detrended residuals, yielding hatλ ± σ_λ; (3) test candidate radices against the measured value.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The multiplicity analysis quantifies the cost of the discovery instrument: the radix-agnostic scan over the Planck multipole range carries $N_{\rm eff} \approx 3000$ independent bins, so a nominal single-bin $p = 10^{-4}$ becomes $p_{\rm global} = 0.26$ after Sidak correction.

   from 4. Discussion

How it is done

1. This work formalizes the radix-agnostic protocol, adds a mandatory certification layer, and demonstrates both a synthetic recovery and a definitive null on real cosmological data.

   from 1. Introduction

2. Given data $(x_i, y_i)$ with $x_i \gt 0$, set $u_i = \ln x_i$ and $f_i = \ln y_i$ (the protocol operates in log-space, where a DSI modulation is sinusoidal).

   from 2.1 The three-stage protocol

3. For non-power-law data (e.g. a spectrum with physical structure such as the $\Lambda$CDM acoustic peaks), the pure-noise bootstrap null is mis-specified: shuffling destroys real, non-DSI structure and inflates significance.

   from 2.3 Model-subtraction mode (G4)

4. The protocol is implemented in a public script with a built-in self-test; all numerical claims below were reproduced with fresh random seeds in the same session.

   from 2.4 Reproducibility

What would falsify it

1. Its observable fingerprint is a log-periodic modulation: in the coordinate $u = \ln x$, a DSI quantity oscillates with angular frequency $$\omega_0 = \frac{2\pi}{\ln\lambda},$$ so that the radix $\lambda$ is encoded in the frequency of a logarithmic-domain oscillation.

   from 1. Introduction

2. If a theory predicts a specific radix — for example, the p-adic hypothesis, under which spacetime structure is valued over $\mathbb{Q}_p$ and DSI inherits the prime radix $\lambda = p$ [; ] — a radix-locked search fixes $\omega_p = 2\pi/\ln p$ and scans only those frequencies.

   from 1. Introduction

3. The null result is consistent with, and extends, the earlier radix-locked analyses: no log-periodic signature is found in the Planck 2018 temperature spectrum beyond the $\Lambda$CDM expectation.

   from 4. Discussion

Where the claim stops

1. The method turns the radix hypothesis from an assumption into a measurement, and is ready for application to other complex systems where DSI may appear.

   from 5. Conclusion

2. But when the radix is unknown — or should not be assumed — the search must be radix-agnostic: estimate $\omega$ continuously and let $\lambda$ emerge as a measured quantity.

   from 1. Introduction

3. A detection claim is certified only when all of the following hold: Bootstrap null (max-statistic).

   from 2.2 Certification

Where to read

1. 1. Introduction277 words · 1 min
2. 2. Methods418 words · 2 min
3. 3. Results333 words · 1 min
4. 4. Discussion187 words · 1 min
5. 5. Conclusion96 words · 1 min
6. Declarations84 words · 1 min