Does the Harmonic Oscillator Attract in Every Number System?

In the paper's words

The universal infrared (IR) attractor claim for the quantum harmonic oscillator (HO) has been examined only over the Archimedean absolute value on ℚ. This work asks whether the attractor status survives under the non‑Archimedean p‑adic and adelic norms. We construct a place‑resolved renormalization‑group (RG) framework, define a dimensionless distance α_v from a flowing bosonic system to the HO fixed point for each place v, and adopt a geometric p‑adic spectrum E^(p)_n=E₀ p^-n organized on Bruhat–Tits tree levels.

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

Numbers it reports

1. Explicit derivations include (i) verification of the adelic product formula for three rationals, (ii) counting of Bruhat–Tits tree vertices up to depth 5, (iii) comparison of constant Archimedean gaps with geometric $p$‑adic gaps for $p=2,3,5$, (iv) exponential RG decay $\alpha_v(\ell)=e^{-\gamma_v\ell}$ with illustrative $\gamma_v=0.1$, (v) testing a product‑of‑distances linkage hypothesis, and (vi) a cross‑place gap‑ratio divergence.

   from Abstract

2. An exact spectral contrast: constant Archimedean gaps versus geometric $p$‑adic gaps with ratio $p$, computed for $p=2,3,5$ (Section 4, A3).

   from 1. Introduction

3. For $p=2$ the $(p+1)$‑regular tree has branching factor $3$. $$ \begin{aligned} N_1 &=3,\\ N_2 &=3\times2=6,\\ N_3 &=3\times4=12,\\ N_4 &=3\times8=24,\\ N_5 &=3\times16=48, \end{aligned} $$ cumulative count $$ S_2(5)=1+3+6+12+24+48=94. $$ Closed form $S_p(N)=1+(p+1)\frac{p^{N}-1}{p-1}$ gives $$ S_2(5)=1+3\frac{2^{5}-1}{1}=1+3\times31=94. $$ For $p=3$ the branching factor is $4$. $$ \begin{aligned} N_1 &=4,\\ N_2 &=4\times3=12,\\ N_3 &=4\times9=36,\\ N_4 &=4\times27=108,\\ N_5 &=4\times81=324, \end{aligned} $$ cumulative count $$ S_3(5)=1+4+12+36+108+324=485. $$ Closed form $ S_3(5)=1+4\frac{3^{5}-1}{2}=1+4\times121=485. ] A3.

   from 4. Analysis

4. Archimedean gaps: $$ \Delta E^{(\infty)}_n = 1\quad\forall n. $$ $p=2$ gaps: $$ \Delta E^{(2)}_0 = \tfrac12,\; \Delta E^{(2)}_1 = \tfrac14,\; \Delta E^{(2)}_2 = \tfrac18,\; \Delta E^{(2)}_3 = \tfrac1{16},\; \Delta E^{(2)}_4 = \tfrac1{32}, $$ ratio $2$ between successive gaps. $p=3$ gaps: $$ \Delta E^{(3)}_0 = \tfrac23,\; \Delta E^{(3)}_1 = \tfrac2{9},\; \Delta E^{(3)}_2 = \tfrac2{27},\; \Delta E^{(3)}_3 = \tfrac2{81}, $$ ratio $3$. $p=5$ gaps: $$ \Delta E^{(5)}_0 = \tfrac45,\; \Delta E^{(5)}_1 = \tfrac4{25},\; \Delta E^{(5)}_2 = \tfrac4{125},\; \Delta E^{(5)}_3 = \tfrac4{625}, $$ ratio $5$.

   from 4. Analysis

5. With $\hbar\omega=E_0=1$, the ratio of the Archimedean gap to a $p$‑adic gap at level $n$ is $$ \frac{\Delta E^{(\infty)}_n}{\Delta E^{(p)}_n} = \frac{1}{p^{-n}\!\left(1-\frac1p\right)} = \frac{p^{n}}{1-\frac1p} = \frac{p^{n+1}}{p-1}. $$ For $p=2$ and $n=5$, $$ \frac{\Delta E^{(\infty)}_5}{\Delta E^{(2)}_5} = \frac{2^{6}}{2-1}=64. $$

   from 4. Analysis

6. Under $E^{(p)}_n=E_0\,p^{-n}$, the $p$‑adic gap ratio equals $p$ for $p=2,3,5$ (A3), distinct from the constant Archimedean gap.

   from 5. Results

How it is done

1. We construct a place‑resolved renormalization‑group (RG) framework, define a dimensionless distance $\alpha_v$ from a flowing bosonic system to the HO fixed point for each place $v$, and adopt a geometric $p$‑adic spectrum $E^{(p)}_n=E_0\,p^{-n}$ organized on Bruhat–Tits tree levels.

   from Abstract

2. We assume the standard classification of absolute values on $\mathbb{Q}$ (Ostrowski’s theorem), whereby $\mathbb{Q}$ admits the usual real absolute value and a family of $p$‑adic absolute values $|\cdot|_p$.

   from 1. Introduction

3. We test whether the adelic product formula forces such a relation, and we identify separately the quantity on which the product formula does bind: the pointwise values of a wavefunction at rational arguments, not its eigenvalues.

   from 3.4 Adelic linkage test

4. All numbers below are derived from the stated inputs; no external data are used.

   from 4. Analysis

What would falsify it

1. An exact $p$‑adic spectrum with constant gaps would falsify R2 and R7.

   from 6.7. Limitations, failure modes and open questions

2. A derivation forcing $\sum_v\gamma_v=0$ from adelic dynamics would falsify R5 and revive a strong place‑invariance claim.

   from 6.7. Limitations, failure modes and open questions

3. Falsification criterion: if the computed ratio deviates from $p$ beyond numerical tolerance (e.g., $|\,\Delta E^{(p)}_n/\Delta E^{(p)}_{n+1}-p\,|\gt 10^{-3}$), then the geometric ladder model and the structural‑transformation verdict of Section 5 (R2, R5, R6) are falsified.

   from 7. Practical Implications and Engineering Pathways

Where the claim stops

1. The $p$‑adic level formula $E^{(p)}_n=E_0\,p^{-n}$ is a modeling assumption; consequently, all quantitative statements that depend on this ladder (R2, R3, R7) inherit this status.

   from 6.7. Limitations, failure modes and open questions

2. An open question is whether a universal mass scale for bosonic fields admits a place‑resolved version.

   from 6.7. Limitations, failure modes and open questions

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 spectrally distinct fixed points—constant gaps in $\mathbb{R}$ versus geometric gaps in ℚ$_{p}$ organized on Bruhat–Tits tree levels.

   from 10. Conclusion

Where to read

1. Abstract181 words · 1 min
2. 1. Introduction301 words · 1 min
3. 2. Background and Related Work59 words · 1 min
4. 3. Methods278 words · 1 min
5. 4. Analysis383 words · 2 min
6. 5. Results125 words · 1 min
7. 6. Discussion1,088 words · 5 min
8. 7. Practical Implications and Engineering Pathways482 words · 2 min
9. 10. Conclusion118 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report159 words · 1 min
12. Appendix B. Claim attribution124 words · 1 min