L‑Function Zeros as Gauge‑Invariant Spectra Across Archimedean and $p$‑adic Geometries

In the paper's words

The quest for a unified spectral invariant that simultaneously governs harmonic resonance on the hyperbolic plane and on p‑adic Bruhat–Tits trees remains open. Existing analogies between Maass‑form eigenvalues of the Laplacian on the modular surface and spectra of discrete Laplacians on p‑adic buildings lack a concrete, gauge‑invariant bridge. We formulate a gauge‑principle hypothesis: the non‑trivial zeros of an L‑function L(s,θ) constitute the universal spectral core, invariant under the choice of Archimedean or non‑Archimedean completion K=C or K=Q_p.

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

Numbers it reports

1. Using the values $\sqrt{2}=1.41421356$ and $\sqrt{5}=2.23606798$ (standard arithmetic), we compute: $p$ $(p+1)$ $2\sqrt{p}$ Lower bound $L$ Upper bound $U$ Width $U-L$ 2 3 $2\cdot1.41421356=2.82842712$ $3-2.82842712=0.17157288$ $3+2.82842712=5.82842712$ $5.65685424$ 5 6 $2\cdot2.23606798=4.47213595$ $6-4.47213595=1.52786405$ $6+4.47213595=10.47213595$ $8.94427190$ All arithmetic follows directly from (4.4) and the square‑root values quoted in [9].

   from 4.5. Numerical illustration for $p=2$ and $p=5$

2. Construct a finite truncation of $\mathcal{T}_p$ for $p=2$ with radius $R=10^{4}$ vertices, compute the spectrum of the combinatorial Laplacian numerically, and examine whether any eigenvalue lies strictly below the lower bound $L=0.17157288$ (for $p=2$) while the associated eigenvector exhibits decay faster than $p^{-n/2}$.

   from (c) Smallest next experiment

3. Under the assumption of access to a high‑performance computing cluster, we project that a truncation experiment with radius $R=10^{4}$ would cost approximately $5\times10^{3}$ compute‑time credits and achieve eigenvalue spectra with absolute error on the order of $10^{-6}$, which would be sufficient to test the lower bound $L$ for $p=2$.

   from (e) Projected cost and performance (projection)

How it is done

1. By constructing an adelic‑conditioned Laplacian $\Delta_{\mathrm{adel}}$ on the infinite $(p+1)$‑regular tree $\mathcal{T}_p$ and comparing its spectrum with the continuous spectrum of the hyperbolic Laplacian, we derive two exact results.

   from Abstract

2. If the conjecture that $L$‑function zeros are the universal spectral invariants holds, then (i) the same set of numbers would dictate eigenvalue spacings of the hyperbolic Laplacian and of the tree Laplacian, and (ii) a single computational experiment could falsify or support a broad class of cross‑domain theories.

   from 1. Introduction

3. These arithmetic identities underpin the $p$‑adic analytic continuation of $L$‑functions that we compare with spectral data. [2] proves that the Fourier coefficients of Siegel‑Eisenstein series admit $p$‑adic meromorphic continuation via the reciprocal of a product of $L$‑functions, explicitly using the identity $\zeta(1-k)^{-1}(1-p^{k-1})^{-1}= \dfrac{U^{*}_{\theta_{k},c}((1+p)^{k-1}-1)(1-c^{k})}{P_{\theta_{k},c}((1+p)^{k-1}-1)}$.

   from 2. Background and Related Work

4. We adopt a purely analytic framework that avoids any numerical simulation beyond the illustrative calculations in Section 5.

   from 3. Methods

What would falsify it

1. If a numerical computation on a sufficiently large finite truncation of $\mathcal{T}_p$ reveals an eigenvalue $\lambda$ whose associated eigenfunction remains normalizable in the limit $N\to\infty$, then the decay argument fails and the gauge‑principle hypothesis is falsified.

   from 6.2. Falsification criterion

2. These findings provide a falsifiable criterion: any observed $\ell^2$ eigenvalue on $\mathcal{T}_p$ would refute the gauge‑principle conjecture.

   from Abstract

3. In high‑energy physics, the inability to predict particle‑mass hierarchies from first principles incurs a cost measured in billions of dollars of collider operation and in the stagnation of model‑building progress.

   from 1. Introduction

Where the claim stops

1. Assumption Sensitivity Infinite regular tree $\mathcal{T}_p$ Truncating the tree (finite graphs) introduces spurious discrete eigenvalues; our result applies only to the limit $N\to\infty$.

   from 6.1. Limitations

2. Standard combinatorial Laplacian $\Delta=(p+1)I-A$ Alternative weightings of edges would modify (4.4); the present analysis assumes unit edge weights.

   from 6.1. Limitations

3. Radial ansatz $f_{n}=c\alpha^{n}$ Non‑radial eigenfunctions could in principle evade the decay argument; however, the tree’s transitive symmetry forces any $\ell^{2}$ eigenfunction to be radial (see standard spectral theory of regular trees).

   from 6.1. Limitations

Where to read

1. Abstract181 words · 1 min
2. 1. Introduction220 words · 1 min
3. 2. Background and Related Work432 words · 2 min
4. 3. Methods241 words · 1 min
5. 4. Analysis309 words · 1 min
6. 5. Results101 words · 1 min
7. 6. Discussion198 words · 1 min
8. 7. Practical Implications and Engineering Pathways281 words · 1 min
9. 8. Conclusion107 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report21 words · 1 min
12. Appendix B. Claim attribution133 words · 1 min