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

Rowan Brad Quni-Gudzinas

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 $f_{\mathrm{NL}}^{(p)} = f_{\mathrm{NL}}^{(0)}\left[1+\varepsilon_p\cos(\omega_p \ln K+\phi)\right]$ 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 $\varepsilon_p \lt 2.5$ 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. 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

\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 $p$. 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. 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.

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. 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 $f_{\mathrm{NL}}(k_1,k_2,k_3)$ 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 $k_i \to p\,k_i$, the reduced bispectrum acquires a multiplicative modulation that is periodic in the logarithm of the overall momentum scale $K = k_1+k_2+k_3$, with the radix-locked frequency $\omega_p$:

f_{\mathrm{NL}}^{(p)}(k_1,k_2,k_3) = f_{\mathrm{NL}}^{(0)}(k_1,k_2,k_3) \left[1 + \varepsilon_p \cos\left(\omega_p \ln K + \phi_p\right)\right], \qquad \omega_p = \frac{2\pi}{\ln 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 $\varepsilon_p$ and the phase $\phi_p$; 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.

#2.2 The trispectrum extension

The same ansatz extends to the four-point function: the trispectrum amplitude $\tau_{\mathrm{NL}}$ carries the modulation

\tau_{\mathrm{NL}}^{(p)} = \tau_{\mathrm{NL}}^{(0)} \left[1 + \varepsilon_p^{(4)} \cos\left(\omega_p \ln K_4 + \phi_p^{(4)}\right)\right], \qquad K_4 = k_1+k_2+k_3+k_4.

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

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.
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.
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 $k_1\le k_2\le k_3$, $k_3\le k_1+k_2$, 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, $\Delta\omega \approx 2\pi/\ln(k_{\max}/k_{\min})$. For the Planck multipole range ($\ell\in[2,2508]$) this gives $\Delta\omega = 1.3644$.

#3.2 Radix separability

The radix frequencies are $\omega_2 = 9.06$, $\omega_3 = 5.72$, $\omega_5 = 3.90$, $\omega_7 = 3.23$. Their pairwise separations are:

Pair	Separation	Resolvable at $\Delta\omega=1.3644$?
2--3	3.35	Yes
2--5	5.16	Yes
2--7	5.84	Yes
3--5	1.82	Yes
3--7	2.49	Yes
5--7	0.68	No

The shape-correlation matrix $C(S_p,S_q)$ 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 $\lvert C\rvert \gt 0.7$ or separation below the frequency resolution):

$p=2$ is cleanly orthogonal to every other small prime ($\lvert C\rvert \le 0.33$, separations $\gt 3$). 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 $0.68 \lt \Delta\omega$): 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. 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 $A_{\mathrm{LPO}} \lt 3\times10^{-3}$ 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). The expected modulation signal is $\varepsilon_p \cdot f_{\mathrm{NL}}^{(0)} \sim 3\times10^{-3}$, 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 (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 ($\omega_2=9.06$, $\omega_3=5.72$, $\omega_5=3.90$, $\omega_7=3.23$) 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:

\boxed{\ \varepsilon_p \lt 2.5 \quad (95\%\ \mathrm{CL},\ \mathrm{T+E},\ \mathrm{SMICA})\ }

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 $\varepsilon_p \lt 1.7$ and the $K\sin$ row $\varepsilon_p \lt 2.3$ at 95% CL.

#5.3 Amplitude-consistency comparison

Constraint	Bound on $\varepsilon_p$	Source
Two-point null (single-modulation)	$\lt 3\times10^{-3}$
Planck 2018 bispectrum (equil-feature)	$\lt 2.5$	this work
Ratio (bispectrum / two-point)	$\approx 830$	—

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

Frequency resolution: separating $p=5$ from $p=7$ requires $\Delta\omega \lt 0.68$, i.e. a log-dynamic range $\ln(k_{\max}/k_{\min}) \gt 9.3$ decades — beyond a single CMB survey but reachable by combining CMB and large-scale-structure bispectra.
Amplitude sensitivity: reaching $\varepsilon_p \sim 0.05$ requires $\sigma(f_{\mathrm{NL}}) \lesssim 1$; reaching the two-point-implied $\varepsilon_p \sim 3\times10^{-3}$ requires $\sigma(f_{\mathrm{NL}}) \sim 10^{-2}$, likely beyond CMB-S4 without an amplification mechanism.
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 and the p-adic CFT formalism provide the theoretical context in which these bounds are interpreted. The tree-like structure of eternal inflation 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: $\varepsilon_p \lt 2.5$ 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 ($A_{\mathrm{LPO}} \lt 3\times10^{-3}$), 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

Funding: No external funding was received for this work.
Competing interests: The author declares no competing interests.
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 https://papers.qnfo.org/papers.
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.
Author contributions: The author conceived the study, performed the analysis, and wrote the manuscript.
Ethics approval: Not applicable.
Consent for publication: Not applicable.
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.
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