Searching for P-Adic Log-Periodic Signatures in the Cosmic Microwave Background Bispectrum: Upper Bounds from Planck 2018

In the paper's words

We search for p-adic (ultrametric) log-periodic signatures in the cosmic microwave background bispectrum. Radix-locked oscillations with angular frequency omega_p = 2*pi/ln(p), p in {2,3,5,7}. Upper bounds from Planck 2018: epsilon_p < 2.5 at 95% CL, no detection; within single-modulation model this is ~830x weaker than the two-point bound (A_LPO < 3e-3).

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

Numbers it reports

1. An empirical search of the Planck 2018 temperature power spectrum placed the first bound on this class of models: the modulation amplitude satisfies $A_{\mathrm{LPO}} \lt 3\times10^{-3}$ at 95% CL for all candidate primes, with log-Bayes factors of $-5.1$ to $-6.5$ against the modulated model.

   from 1. Introduction

2. The high-frequency log-oscillatory families bound even more tightly: the $K^2\cos$ row gives $\varepsilon_p \lt 1.7$ and the $K\sin$ row $\varepsilon_p \lt 2.3$ at 95% CL.

   from 5.2 Radix-locked upper bounds

3. Throughout, $\varepsilon_p$ denotes the dimensionless amplitude of the log-periodic modulation in the reduced bispectrum, and all limits are 95% CL unless stated.

   from 1. Introduction

4. The claim tested here has three concrete falsification conditions, fixed before the analysis D1 (no modulation): Planck 2018 shows no log-periodic modulation at any radix-locked $\omega_p$ at a sensitivity that bounds $\varepsilon_p$ at 95% CL with a look-elsewhere correction.

   from 2.3 Falsifiable content (pre-registered)

5. The constraint set is taken from the public Planck 2018 results paper on primordial non-Gaussianity (arXiv:1905.05697), whose abstract and body tables were retrieved and verified live for this analysis: Base shapes (68% CL, T+E): $f_{\mathrm{NL}}^{\mathrm{local}} = -0.9\pm5.1$, $f_{\mathrm{NL}}^{\mathrm{equil}} = -26\pm47$, $f_{\mathrm{NL}}^{\mathrm{ortho}} = -38\pm24$.

   from 5.1 Data and verification

6. Feature/resonance (95% CL, SMICA, T+E): constant $2.5$; equilateral $2.5$; flattened $2.4$; $K^2\cos$ $1.7$; $K\sin$ $2.3$.

   from 5.1 Data and verification

How it is done

1. An empirical search of the Planck 2018 temperature power spectrum placed the first bound on this class of models: the modulation amplitude satisfies $A_{\mathrm{LPO}} \lt 3\times10^{-3}$ at 95% CL for all candidate primes, with log-Bayes factors of $-5.1$ to $-6.5$ against the modulated model.

   from 1. Introduction

2. To assess whether a radix-locked detection could be identified, and distinguished from the resonant-feature family, we compute the shape-space correlation between templates over the tetrahedral momentum domain $k_1\le k_2\le k_3$, $k_3\le k_1+k_2$, with log-uniform sampling and weight $1/(k_1k_2k_3)$ (the scale-invariant measure): $$C(S_a,S_b) = \frac{\sum_i w_i S_a(k_i)\,S_b(k_i)} {\sqrt{\sum_i w_i S_a(k_i)^2}\sqrt{\sum_i w_i S_b(k_i)^2}}.$$ The frequency resolution is set by the log-dynamic range of the data, $\Delta\omega \approx 2\pi/\ln(k_{\max}/k_{\min})$.

   from 3.1 Shape correlator

3. This partial identifiability is a bound on the framework's own testability and is reported as such (D3): no claimed radix identification between $(3,5)$ or $(5,7)$ at Planck resolution can be trusted without a wider log-dynamic range.

   from 3.2 Radix separability

4. The two-point bound $A_{\mathrm{LPO}} \lt 3\times10^{-3}$ then implies, in a single-modulation model, $$\varepsilon_p \lesssim A_{\mathrm{LPO}} \approx 3\times10^{-3}.$$ The Planck 2018 sensitivity to an equilateral-type resonant shape is $\sigma(f_{\mathrm{NL}}) \sim \mathcal{O}(10)$ in the $f_{\mathrm{NL}}$ normalization (see Section 5).

   from 4. Amplitude consistency with the two-point null

What would falsify it

1. The claim tested here has three concrete falsification conditions, fixed before the analysis D1 (no modulation): Planck 2018 shows no log-periodic modulation at any radix-locked $\omega_p$ at a sensitivity that bounds $\varepsilon_p$ at 95% CL with a look-elsewhere correction.

   from 2.3 Falsifiable content (pre-registered)

2. The concrete prediction for the CMB angular power spectrum takes the form $$\ell(\ell+1)C_\ell = A\left(\frac{\ell}{\ell_0}\right)^{1-n_s} \left[1 + B\cos\left(\frac{2\pi}{\ln p}\ln\frac{\ell}{\ell_0}+\phi\right)\right],$$ i.e. a log-periodic modulation with angular frequency $\omega_p = 2\pi/\ln p$ locked to a prime radix $p$.

   from 1. Introduction

3. This is a genuine null result: the simplest two-point signature of p-adic structure is absent at the sensitivity of Planck.

   from 1. Introduction

Where the claim stops

1. Within the single-modulation model the bispectrum bound is ~830 times weaker than the two-point bound ($A_{\mathrm{LPO}} \lt 3\times10^{-3}$), confirming that the higher-order channel does not amplify the p-adic signal.

   from 7. Conclusion

2. A shared radix frequency across channels is itself a falsifiable prediction: the claim is disconfirmed if the best-fit bispectrum and trispectrum frequencies disagree beyond their combined uncertainty.

   from 2.2 The trispectrum extension

3. The $(3,5)$ pair is degenerate ($C=-0.77$, anti-correlated): a signal in this band cannot be uniquely attributed to $p=3$ or $p=5$ from shape alone.

   from 3.2 Radix separability

Where to read

1. 1. Introduction329 words · 1 min
2. 2. The p-adic log-periodic bispectrum template305 words · 1 min
3. 3. Template identifiability in shape space382 words · 2 min
4. 4. Amplitude consistency with the two-point null142 words · 1 min
5. 5. Constraints from Planck 2018287 words · 1 min
6. 6. Discussion229 words · 1 min
7. 7. Conclusion167 words · 1 min
8. Declarations136 words · 1 min