Is the Harmonic Oscillator an Infrared Attractor in All Places? A Place-Resolved Analysis of the Harmonic Paradigm

In the paper's words

The conjecture that the harmonic oscillator is a universal infrared (IR) attractor for bosonic quantum systems has so far been formulated and tested only over the Archimedean place, i.e., with respect to the usual real absolute value. We ask whether this attractor status survives a change of place in the sense of Ostrowski's classification of absolute values on Q. We set up a place-resolved renormalization-group (RG) framework in which a dimensionless distance α_v from a flowing bosonic system to the oscillator fixed point is defined separately for the Archimedean place v=∞ and for the p-adic places v=p, using a putative p-adic fixed point whose spectrum we model on the levels of the Bruhat–Tits tree of Q_p (an explicit modeling assumption of this paper; the adelic oscillator of [8] is established qualitatively, but its supplied summary states no level formula). [...]

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

Numbers it reports

1. An internal red-team assessment describes the Paradigm's ambitious scope, including an "8-rung ladder spanning from transmon anharmony to quantum gravity" [14], and the Paradigm has practical consequences: if the HO is the IR attractor of quantum mechanics, then bosonic encodings are argued to be the native encoding of quantum information, a claim backed by a resource comparison in which bosonic codes require 5–40 times fewer photons and roughly 100 times fewer modes than surface codes at a target logical error rate of $p_L=10^{-6}$ [12].

   from 1. Introduction

2. Archimedean: $\Delta E^{(\infty)}_n=\hbar\omega$ for all $n$; with $\hbar\omega$ normalized to $1$, gaps are $(1,1,1,\ldots)$. $p$-adic (model of Section 3.2, $E_0=1$, $p=2$): $$\Delta E^{(2)}_0=1-\tfrac{1}{2}=\tfrac{1}{2},\quad \Delta E^{(2)}_1=\tfrac{1}{2}-\tfrac{1}{4}=\tfrac{1}{4},\quad \Delta E^{(2)}_2=\tfrac{1}{8},\quad \Delta E^{(2)}_3=\tfrac{1}{16}.$$ Gap ratio check: $\Delta E^{(2)}_0/\Delta E^{(2)}_1=(1/2)/(1/4)=2=p$, and $\Delta E^{(2)}_1/\Delta E^{(2)}_2=(1/4)/(1/8)=2=p$.

   from 4. Analysis

3. For $p=3$, $E_0=1$: gaps are $1-1/3=2/3$, then $1/3-1/9=2/9$, then $2/27$; ratio $(2/3)/(2/9)=3=p$.

   from 4. Analysis

4. Each successive interval of $20$ RG steps reduces the distance by the factor $e^{-2}\approx 0.135335$, i.e., by roughly $86.466\%$.

   from 4. Analysis

5. Under the stated geometric model $E^{(p)}_n=E_0\,p^{-n}$, the $p$-adic oscillator has gap ratio exactly $p$ between adjacent levels, computed for $p=2$ (gaps $1/2,1/4,1/8,1/16$) and $p=3$ (gaps $2/3,2/9,2/27$), against the constant Archimedean gap $\hbar\omega$ (A3).

   from 5. Results

How it is done

1. We derive explicitly: (i) the Archimedean level structure $E_n=\hbar\omega(n+\tfrac{1}{2})$ with constant spacing; (ii) a geometric $p$-adic level ladder $E^{(p)}_n=E_0\,p^{-n}$ with adjacent-gap ratio $p$, under a clearly labeled modeling assumption; (iii) an exponential RG approach $\alpha_v(\ell)=e^{-\gamma_v\ell}$, evaluated numerically; and (iv) a test of adelic linkage via the product formula, verified exactly for $x=2/3$.

   from Abstract

2. By Ostrowski's theorem, the nontrivial absolute values on $\mathbb{Q}$ are, up to equivalence, the usual real absolute value and the $p$-adic absolute values $|\cdot|_p$.

   from 1. Introduction

3. The study of what attracts to attractors in [9] examines, for systems relevant to heavy-ion collisions, whether, how, and to what extent solutions become insensitive to aspects of their initial conditions, in Israel–Stewart theory and kinetic theory where a universal attractor solution governs the approach.

   from 2. Background and Related Work

4. We test whether any such identity is forced by the adelic structure, using the product formula as the only adelic input, and we also identify where the product formula does bind (amplitudes).

   from 3.4 Adelic linkage test

What would falsify it

1. An internal red-team assessment describes the Paradigm's ambitious scope, including an "8-rung ladder spanning from transmon anharmony to quantum gravity" [14], and the Paradigm has practical consequences: if the HO is the IR attractor of quantum mechanics, then bosonic encodings are argued to be the native encoding of quantum information, a claim backed by a resource comparison in which bosonic codes require 5–40 times fewer photons and roughly 100 times fewer modes than surface codes at a target logical error rate of $p_L=10^{-6}$ [12].

   from 1. Introduction

2. (i) An exact $p$-adic spectrum with constant gaps would falsify R2.

   from 6. Discussion

3. (ii) A derivation forcing $\sum_v\gamma_v=0$ from adelic dynamics would falsify R5 and revive strong place-invariance.

   from 6. Discussion

Where the claim stops

1. The supplied literature establishes that the adelic oscillator exists and softens the uncertainty relation [8], but its summary does not state the spectrum; our geometric ladder is a principled guess consistent with tree-level organization [13], and every quantitative result that depends on it (R2, and the spectral interpretation of R3) inherits that status.

   from 6. Discussion

2. The warning from [7] that the inverted oscillator involves unclear points in the $\omega\to i\omega$ redefinition suggests that repulsive sectors are genuinely delicate and should not be assumed mirror-symmetric.

   from 6. Discussion

3. Within a place-resolved RG framework, with explicit derivations and clearly labeled modeling assumptions, we find that the oscillator can attract in each place but with structurally different spectra — constant gaps in $\mathbb{R}$, geometric gaps in $\mathbb{Q}_p$ organized on Bruhat–Tits tree levels — and that the adelic product formula constrains amplitudes, not spectra, imposing no linkage between the place-wise attractor distances.

   from 7. Conclusion

Where to read

1. Abstract246 words · 1 min
2. 1. Introduction473 words · 2 min
3. 2. Background and Related Work1,032 words · 4 min
4. 3. Methods398 words · 2 min
5. 4. Analysis531 words · 2 min
6. 5. Results204 words · 1 min
7. 6. Discussion437 words · 2 min
8. 7. Conclusion127 words · 1 min
9. Appendix A. Divergence report443 words · 2 min