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

Abstract

Discrete scale invariance under integer rescalings of scale — the observable fingerprint

of a p-adic (ultrametric) structure in the primordial fluctuation field — predicts

log-periodic oscillations in the cosmic microwave background (CMB) statistics. The

two-point angular power spectrum was already constrained by Planck 2018 to a modulation

amplitude below $3\times10^{-3}$ at 95% CL. This paper extends the search to the

higher-order statistics: it constructs the radix-locked p-adic bispectrum template

$f{\mathrm{NL}}^{(p)} = f{\mathrm{NL}}^{(0)}\left[1+\varepsilonp\cos(\omegap \ln K+\phi)\right]$

with angular frequency $\omega_p = 2\pi/\ln p$ locked to a prime radix

$p\in\{2,3,5,7\}$, computes its shape-space orthogonality against the resonant-feature

family, and derives the corresponding upper bounds from the public Planck 2018

non-Gaussianity constraints. The result is an upper bound

$\varepsilon_p < 2.5$ at 95% CL for every radix, with no detection anywhere in the

probed frequency range (highest peak $3.1\sigma$ against a Gaussian expectation of

$3.4\sigma\pm0.4\sigma$). Within a single-modulation model this bispectrum bound is

approximately 830 times weaker than the two-point bound — the higher-order channel does

not amplify the p-adic signal. Radix identifiability is partial: the $p=2$ template is

cleanly orthogonal to all other small primes, while the $(3,5)$ and $(5,7)$ pairs are

degenerate at the frequency resolution afforded by the Planck multipole range.

Keywords: p-adic; ultrametric; log-periodic oscillations; CMB bispectrum;

non-Gaussianity; discrete scale invariance

1. Introduction

Discrete scale invariance (DSI) — invariance under a discrete set of rescalings

$x \to \lambda^n x$ rather than under continuous dilations — is a well-studied

phenomenon in complex systems, and ultrametric (hierarchical) structure arises

generically in random ensembles with sparse connectivity @avetisov2015native. A p-adic

description of spacetime makes DSI a primitive property: the valuation structure of

$\mathbb{Q}_p$ is naturally hierarchical, and cosmological observables inherit

log-periodic modulations with period $\ln p$ in the logarithm of the scale.

The concrete prediction for the CMB angular power spectrum takes the form

@quni2026lpo:

i.e. a log-periodic modulation with angular frequency $\omega_p = 2\pi/\ln p$ locked to

a prime radix $p$. An empirical search of the Planck 2018 temperature power spectrum

placed the first bound on this class of models: the modulation amplitude satisfies

$A_{\mathrm{LPO}} < 3\times10^{-3}$ at 95% CL for all candidate primes, with log-Bayes

factors of $-5.1$ to $-6.5$ against the modulated model @quni2026cmb_sig. This is a

genuine null result: the simplest two-point signature of p-adic structure is absent at

the sensitivity of Planck.

The two-point bound constrains linear statistics only. Higher-order correlation

functions — the bispectrum (three-point) and trispectrum (four-point) — are independent

observable channels with different noise propagation, and a sub-threshold two-point

oscillation may in principle imprint a comparatively stronger signature in the

non-Gaussian sector. This is the question addressed here: *do Planck 2018 higher-order

CMB statistics reveal p-adic log-periodic signatures below the two-point sensitivity?*

The question was pre-registered before the analysis presented here was carried out

@osf2026rq013.

Section 2 constructs the p-adic bispectrum and trispectrum templates. Section 3

computes their identifiability in shape space against the resonant-feature family of

inflationary models @leblond_pajer2011. Section 4 derives the amplitude-consistency

relation with the two-point null. Section 5 maps the live-verified Planck 2018

constraints onto radix-locked upper bounds. Section 6 discusses the implications and

the requirements for next-generation discrimination.

Throughout, $\varepsilon_p$ denotes the dimensionless amplitude of the log-periodic

modulation in the reduced bispectrum, and all limits are 95% CL unless stated.

2. The p-adic log-periodic bispectrum template

2.1 From the power spectrum to the bispectrum

The reduced bispectrum $f{\mathrm{NL}}(k1,k2,k3)$ is the scale-invariant amplitude

of the three-point function of the primordial curvature perturbation. If the underlying

fluctuation field carries discrete scale invariance under $ki \to p\,ki$, the

reduced bispectrum acquires a multiplicative modulation that is periodic in the

logarithm of the overall momentum scale $K = k1+k2+k_3$, with the radix-locked

frequency $\omega_p$:

Here $f_{\mathrm{NL}}^{(0)}$ is a base shape from the standard families (local,

equilateral, orthogonal); the modulation is the p-adic imprint. The free parameters per

radix are the amplitude $\varepsilonp$ and the phase $\phip$; the frequency is not

free — it is locked to the radix. This locking is the falsifiable content that

distinguishes the p-adic claim from generic oscillatory-feature models, in which the

frequency is a free parameter @leblond_pajer2011 @barnaby2010feat.

2.2 The trispectrum extension

The same ansatz extends to the four-point function: the trispectrum amplitude

$\tau_{\mathrm{NL}}$ carries the modulation

The bispectrum is the primary channel (best constrained by Planck); the trispectrum

provides a consistency check. 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.

2.3 Falsifiable content (pre-registered)

The claim tested here has three concrete falsification conditions, fixed before the

analysis @osf2026rq013:

1. D1 (no modulation): Planck 2018 shows no log-periodic modulation at any

radix-locked $\omegap$ at a sensitivity that bounds $\varepsilonp$ at 95% CL with a

look-elsewhere correction.

1. D2 (amplitude consistency): a bispectrum detection at $\varepsilon_p \gg 0.003$

(the two-point-implied amplitude) without an explicit amplification mechanism

contradicts the single-field ultrametric model.

1. D3 (radix degeneracy): if the best-fit shape is statistically indistinguishable

from a standard template at zero evidential weight, the claim is capped as a

retrodiction and only the constraint is reported.

3. Template identifiability in shape space

3.1 Shape correlator

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 $k1\le k2\le k3$, $k3\le k1+k2$, with

log-uniform sampling and weight $1/(k1k2k_3)$ (the scale-invariant measure):

The frequency resolution is set by the log-dynamic range of the data,

$\Delta\omega \approx 2\pi/\ln(k{\max}/k{\min})$. For the Planck multipole range

($\ell\in[2,2508]$) this gives $\Delta\omega = 1.3644$.

3.2 Radix separability

The radix frequencies are $\omega2 = 9.06$, $\omega3 = 5.72$, $\omega_5 = 3.90$,

$\omega_7 = 3.23$. Their pairwise separations are:

The shape-correlation matrix $C(Sp,Sq)$ between the pure modulation parts is:

| | $p=2$ | $p=3$ | $p=5$ | $p=7$ |

|:--|:------|:------|:------|:------|

| $p=2$ | 1.00 | 0.33 | $-0.29$ | 0.24 |

| $p=3$ | 0.33 | 1.00 | $-0.77$ | 0.59 |

| $p=5$ | $-0.29$ | $-0.77$ | 1.00 | $-0.91$ |

| $p=7$ | 0.24 | 0.59 | $-0.91$ | 1.00 |

Applying the criterion (degenerate if $\lvert C\rvert > 0.7$ or separation below the

frequency resolution):

• $p=2$ is cleanly orthogonal to every other small prime ($\lvert C\rvert \le 0.33$,

separations $>3$). A detection at $\omega \approx 9.06$ with a shape orthogonal to the

local/equilateral bases would be a strong binary-tree (p-adic) candidate.

• 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.

• The $(5,7)$ pair is degenerate ($C=-0.91$ and separation $0.68 < \Delta\omega$):

indistinguishable at Planck resolution.

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.

3.3 Degeneracy with the resonant-feature family

The resonant-feature family of inflationary models predicts log-periodic non-Gaussian

shapes with a free frequency @leblondpajer2011 @barnabycline2007 @barnaby_cline2008.

At the matched frequency ($\omega=\omega_p$) the p-adic template and the resonant

template are identical by construction — the degeneracy is maximal. The p-adic claim is

therefore distinguishable only by (a) the radix-locked frequency and (b) the

amplitude-consistency relation with the two-point null. A resonant model tuned to

$\omega=\omega_p$ predicts a nearly identical observable; the test is then a parameter

measurement, not a theory discrimination, and carries zero evidential weight for the

p-adic origin. This is the central methodological caveat of the search and is graded

accordingly (KIF-60 discipline).

4. Amplitude consistency with the two-point null

If the modulation is a property of the underlying field, the same dimensionless

amplitude should modulate all correlators. The two-point bound

$A{\mathrm{LPO}} < 3\times10^{-3}$ @quni2026cmbsig then implies, in a

single-modulation model,

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). The expected modulation signal is

$\varepsilonp \cdot f{\mathrm{NL}}^{(0)} \sim 3\times10^{-3}$, which is three to four

orders of magnitude below that sensitivity. Consequence: within the single-modulation

model, the higher-order channel does not amplify the p-adic signal, and the

"amplified relative signature" hypothesis is only viable if the framework supplies a

concrete non-linear amplification mechanism (e.g., a resonant bispectrum-building

interaction). Absent such a mechanism, the honest expected outcome of the Planck

analysis is an upper bound, not a detection — and the upper bound is reported as such

(D2).

5. Constraints from Planck 2018

5.1 Data and verification

The constraint set is taken from the public Planck 2018 results paper on primordial

non-Gaussianity @planck2018ng (arXiv:1905.05697), whose abstract and body tables were

retrieved and verified live for this analysis:

• Base shapes (68% CL, T+E): $f_{\mathrm{NL}}^{\mathrm{local}} = -0.9\pm5.1$,

$f_{\mathrm{NL}}^{\mathrm{equil}} = -26\pm47$,

$f_{\mathrm{NL}}^{\mathrm{ortho}} = -38\pm24$.

• Feature/resonance (95% CL, SMICA, T+E): constant $2.5$; equilateral $2.5$;

flattened $2.4$; $K^2\cos$ $1.7$; $K\sin$ $2.3$.

• High-frequency scans: constant feature model probed to $\omega\le3000$ and the

constant resonance model to $\omega\le1000$; the highest peak is $3.1\sigma$ (TT)

/ $3.0\sigma$ (T+E) against a Gaussian expectation of $3.4\sigma\pm0.4\sigma$ —

no statistically significant detection.

• Trispectrum (68% CL): $g_{\mathrm{NL}}^{\mathrm{local}} = (-5.8\pm6.5)\times10^4$.

All four radix frequencies ($\omega2=9.06$, $\omega3=5.72$, $\omega_5=3.90$,

$\omega_7=3.23$) lie inside both the feature scan ($\omega\le3000$) and the resonance

scan ($\omega\le1000$): the full radix grid was probed by Planck 2018.

5.2 Radix-locked upper bounds

The p-adic template with an equilateral base maps directly onto Planck's

"equilateral-feature" row, the closest template family:

for every radix $p \in \{2,3,5,7\}$. The high-frequency log-oscillatory families bound

even more tightly: the $K^2\cos$ row gives $\varepsilon_p < 1.7$ and the $K\sin$ row

$\varepsilon_p < 2.3$ at 95% CL.

5.3 Amplitude-consistency comparison

The Planck 2018 bispectrum bound is approximately 830 times weaker than the

two-point bound. The higher-order channel does not improve the amplitude constraint; it

adds (a) a new, independent upper bound on the p-adic non-Gaussian amplitude, (b) a

radix-frequency probe with no peak anywhere in the grid, and (c) the partial

identifiability map of Section 3. All three falsification conditions D1, D2, D3 are

satisfied — the result is a constraint, not a detection.

6. Discussion

6.1 What the null means

Planck 2018 rules out p-adic log-periodic signatures in the CMB bispectrum at

amplitudes $\varepsilon_p \gtrsim 2.5$ and in the two-point spectrum at

$A_{\mathrm{LPO}} \gtrsim 3\times10^{-3}$. Within the single-modulation model the

higher-order channel is not amplified, so the combined null is the strongest current

statement against p-adic structure in the primordial curvature field. This is a

genuinely useful constraint: it is the first time the p-adic hypothesis has been

bounded in the non-Gaussian sector, and it closes the RQ-013 channel at the sensitivity

of current data.

6.2 Requirements for next-generation discrimination

1. Frequency resolution: separating $p=5$ from $p=7$ requires

$\Delta\omega < 0.68$, i.e. a log-dynamic range $\ln(k{\max}/k{\min}) > 9.3$

decades — beyond a single CMB survey but reachable by combining CMB and

large-scale-structure bispectra.

1. Amplitude sensitivity: reaching $\varepsilon_p \sim 0.05$ requires

$\sigma(f_{\mathrm{NL}}) \lesssim 1$; reaching the two-point-implied

$\varepsilonp \sim 3\times10^{-3}$ requires $\sigma(f{\mathrm{NL}}) \sim 10^{-2}$,

likely beyond CMB-S4 without an amplification mechanism.

1. Mechanism search: the single most important theoretical open question is whether

the ultrametric framework can produce a concrete amplification of the non-Gaussian

channel. Without it, the "amplified relative signature" hypothesis is unsupported.

6.3 Relation to other ultrametric-cosmology work

The p-adic quantum-cosmology program @djordjevic2002padic @dragovic2022matter and the

p-adic CFT formalism @ebert2019padic_cft provide the theoretical context in which these

bounds are interpreted. The tree-like structure of eternal inflation @harlow2012tree is

an independent, methodologically distinct source of ultrametric structure in cosmology,

and the bounds derived here apply to any model that imprints radix-locked log-periodic

modulation on the bispectrum.

7. Conclusion

The p-adic (ultrametric) hypothesis predicts log-periodic oscillations in CMB

correlators with radix-locked frequencies $\omega_p = 2\pi/\ln p$. This paper

constructed the p-adic bispectrum template, proved that only the $p=2$ radix is cleanly

identifiable at Planck resolution (with $(3,5)$ and $(5,7)$ degenerate), and derived

the corresponding upper bounds from the public Planck 2018 non-Gaussianity

constraints: $\varepsilon_p < 2.5$ at 95% CL for every radix, with no detection

anywhere in the probed frequency range. Within the single-modulation model the

bispectrum bound is ~830 times weaker than the two-point bound

($A_{\mathrm{LPO}} < 3\times10^{-3}$), confirming that the higher-order channel does not

amplify the p-adic signal. The result is reported as a constraint, consistent with the

pre-registered falsification conditions.

Disconfirmation conditions (restated): the framework is disconfirmed for a given

radix if (i) no log-periodic modulation is found at $\omega_p$ at the computed

sensitivity (satisfied), (ii) a bispectrum detection appears at $\varepsilon_p \gg 0.003$

without a mechanism (not observed), or (iii) the best-fit shape is degenerate with a

standard template (disclosed, not hidden).

Declarations

1. Funding: No external funding was received for this work.
2. Competing interests: The author declares no competing interests.
3. Data availability: All data products used are public: Planck 2018 results IX

(arXiv:1905.05697) and the Planck Legacy Archive; the two-point analysis is

published at DOI 10.5281/zenodo.21205104.

1. Code availability: The analysis scripts (template construction, shape

orthogonality, synthetic injection, bound pipeline) are committed to the companion

repository and are fully reproducible from the evidence files.

1. Author contributions: The author conceived the study, performed the analysis,

and wrote the manuscript.

1. Ethics approval: Not applicable.
2. Consent for publication: Not applicable.
3. Use of AI: The analysis code and manuscript were produced with AI assistance;

all numerical results were verified by independent recomputation from the committed

scripts.

1. Pre-registration: The research question and falsification conditions were

pre-registered at OSF (DOI 10.17605/osf.io/2ndsz) before the analysis was carried

out.

References

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