Why a Fixed Doubling Rule Fails in Quantum Physics

In the paper's words

Discrete scale invariance (DSI) — invariance under rescaling by a fixed factor rather than by any factor — is established in few-body quantum physics, Dirac semimetals, and renormalization-group limit cycles. A recurring conjecture holds that the scaling factor itself might be universal, with the dyadic value 2 appearing across unrelated domains. This paper tests that conjecture against the published few-body and condensed-matter evidence.

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

Numbers it reports

1. The three-boson Efimov spectrum has scaling factor $22.694$, which we show corresponds to an RG index $s_0 = 2\pi/\ln(22.694) = 2.0124$ and a log-periodic period $\Delta(\ln E) = 3.1222$, i.e. $4.505$ octaves — a factor of $4.505$ larger than the dyadic period $\ln 2 = 0.69315$.

   from Abstract

2. We conclude that DSI is real and broadly shared, but the specific factor-2 universality hypothesis is excluded by existing data at the level of a factor $11.347$ per spectral step.

   from Abstract

3. The three-boson Efimov factor is $22.694$ [4], not $2$; the scaling-trap factor is continuously tunable through the slope of the scattering-length profile [2]; the Dirac-semimetal factors depend on magnetic-field regime and dimensionality [1]; and the log-periodic modulations proposed in quantum-gravity contexts carry an arbitrary period parameter $\omega$ [3].

   from 1. Introduction

4. The contributions of this paper are: A derivation, from the published Efimov scaling factor $22.694$ [4], of the renormalization-group index $s_0 = 2.0127$, the log-periodic period $\Delta(\ln E) = 3.1222$, and the octave count $4.505$, establishing quantitatively how far the confirmed few-body factor lies from the dyadic value (Section 4).

   from 1. Introduction

5. A falsification criterion for universal dyadic periodicity, stated as a measurable period in $\ln E$, together with the observation that factor 2 does appear in analysis conventions (dyadic partitions of frequency space [11]) and must therefore be distinguished from factor 2 as physics (Section 5).

   from 1. Introduction

6. The scaling factor is not arbitrary: for equal-mass bosons the three-boson spectrum has the discrete scaling factor $22.694$, a number large enough that measuring signatures of two consecutive trimer levels is experimentally challenging, since the scattering length must simultaneously exceed the van der Waals length $r_{\text{vdW}}$ and remain below the de Broglie wavelength $\lambda_{\text{dB}}$ [4].

   from 2. Background and Related Work

How it is done

1. Experimental groups measuring few-body bound states in ultracold atoms, transport in Dirac materials, and spectra near charged impurities all face the same analysis problem: when a spectrum repeats itself under rescaling, what is the scaling factor, and does it carry fundamental meaning?

   from 1. Introduction

2. Three identical bosons near a two-body $s$-wave resonance can exhibit an infinite tower of bound states with a geometric energy sequence, reflecting a broken continuous scaling symmetry.

   from 2. Background and Related Work

3. This paper is an analytical reanalysis of published scaling factors; it performs no experiment, measurement, or simulation.

   from 3. Methods

4. Step 1: The Efimov log-period.

   from 4. Analysis

What would falsify it

1. The practical consequence is a sharp falsification criterion: any claimed universal dyadic periodicity in a physical spectrum must survive comparison against the measured Efimov period of $3.1222$ in $\ln E$, which it cannot if the period is $0.69315$.

   from Abstract

2. A falsification criterion for universal dyadic periodicity, stated as a measurable period in $\ln E$, together with the observation that factor 2 does appear in analysis conventions (dyadic partitions of frequency space [11]) and must therefore be distinguished from factor 2 as physics (Section 5).

   from 1. Introduction

3. This distinction — factor 2 as the base of a chosen decomposition versus factor 2 as a physical period — is exactly the confusion the factor-2 universality conjecture risks, and it motivates the falsification criterion of Section 5.

   from 2. Background and Related Work

Where the claim stops

1. By extracting the empirically measured Efimov scaling factor $22.694$ and converting it to a log‑periodic period of $3.1222$, we have shown that the universal‑dyadic hypothesis fails by a factor of $4.505$ in period and $11.347$ per spectral step.

   from 8. Conclusion

2. The practical consequence is a sharp falsification criterion: any claimed universal dyadic periodicity in a physical spectrum must survive comparison against the measured Efimov period of $3.1222$ in $\ln E$, which it cannot if the period is $0.69315$.

   from Abstract

3. The broader intuition behind the factor-2 conjecture — that discrete scaling loops can replace continuous scale invariance — has solid empirical support; the specific universal value does not.

   from 1. Introduction

Where to read

1. Abstract236 words · 1 min
2. 1. Introduction545 words · 2 min
3. 2. Background and Related Work1,448 words · 6 min
4. 3. Methods357 words · 2 min
5. 4. Analysis750 words · 3 min
6. 5. Results332 words · 1 min
7. 6. Discussion81 words · 1 min
8. 7. Practical Implications and Engineering Pathways410 words · 2 min
9. 8. Conclusion123 words · 1 min
10. Provenance and AI Involvement225 words · 1 min