L‑Functions as Gauge‑Invariant Spectra Across Tree, Hyperbolic, and $p$‑adic Geometries

In the paper's words

Discrete hierarchical substrates such as Bruhat–Tits trees, p‑adic lattices, and prime‑factorization hierarchies generate continuous geometric phenomena without a transparent computational bridge. Existing p‑adic L‑function constructions, Bernoulli‑number expansions, and tree‑based phase‑transition criteria provide concrete arithmetic data that can be compared across these geometries. This work treats the choice of field mathbb R versus mathbb Q_p as a gauge choice and asks which spectral features survive.

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

Numbers it reports

1. Therefore $$ -\zeta(-1)^{-1}=12. $$ Hence the product $\zeta(-1)^{-1}(1-p^{k-1})^{-1}$ evaluates to $12$ for the chosen parameters, while the factor $(1-p^{k-1})^{-1}$ alone equals $-1$.

   from 4.5 Reciprocal product $(1-p^{k-1})^{-1}$ for $p=2$, $k=2$ from [1]

2. Using the Euler product for $\zeta(2)=\pi^{2}/6\approx1.644934$ and a partial sum of $L(2,\chi_{-4})=\sum_{n\ge1}\chi_{-4}(n)/n^{2}$ truncated after $N=10$ terms, we list the first ten non‑zero terms: $$ \frac{1}{1^{2}}\; -\;\frac{1}{3^{2}}\; +\;\frac{1}{5^{2}}\; -\;\frac{1}{7^{2}}\; +\;\frac{1}{9^{2}}\;=\;1-0.111111+0.04-0.020408+0.012345=0.920826. $$ The alternating‑series error bound tells us that the true value lies between $0.920826-\frac{1}{11^{2}}\approx0.912562$ and $0.920826$.

   from 4.6 Theta‑function value $L(2,\vartheta)$ interval from [1] and [8]

3. Using the bound $1/11^{2}=0.008264$, we obtain $L(2,\chi_{-4})\in[0.920826-0.008264,\,0.920826]=[0.912562,0.920826]$.

   from 4.6 Theta‑function value $L(2,\vartheta)$ interval from [1] and [8]

4. A prototype computation of $\vartheta(q)$ for all $b\le20$ requires roughly $10^{3}$ CPU‑hours on a standard high‑performance node (projection based on the $O(b^{2})$ evaluation cost of theta series).

   from 7. Practical Implications and Engineering Pathways

How it is done

1. We derive explicit arithmetic illustrations: the Bernoulli numbers $B_{2}=1/6$ and $B_{10}=5/66$, the $p$‑adic norm $|3^{2}|_{3}=1/9$, the Kummer congruence $B_{10}/10\equiv B_{6}/6\pmod5$, the phase‑transition condition $p=3$ for a three‑state Potts model, the product $\zeta(-1)^{-1}(1-p^{k-1})^{-1}=12$ for $p=2$, $k=2$ (while $(1-p^{k-1})^{-1}=-1$), the theta‑function interval $L(2,\vartheta)\in[1.50111,1.51568]$, the root‑of‑unity theta sum $|\sum_{n\ge1}2^{-n}i^{n^{2}}|=\sqrt5/3\approx0.745356$, the tree spectral edge $2\sqrt{p}$ (e.g. $2.8284271$ for $p=2$), and a cophenetic‑distance counterexample $d(4,9)=4\gt \max\{d(4,6),d(6,9)\}=2.

   from Abstract

2. Identification of a minimal computational experiment that can falsify the conjecture by comparing theta‑function interference peaks with Farey fractions (Section 7).

   from 1. Introduction

3. Theorem 2.1 of [3] expresses $L_{p,r}$ at non‑positive integers as finite sums of twisted multiple Bernoulli numbers ${\mathfrak B}$, providing explicit combinatorial formulas for spectral coefficients.

   from 2. Background and Related Work

4. We adopt a three‑step methodological framework.

   from 3. Methods

What would falsify it

1. We derive explicit arithmetic illustrations: the Bernoulli numbers $B_{2}=1/6$ and $B_{10}=5/66$, the $p$‑adic norm $|3^{2}|_{3}=1/9$, the Kummer congruence $B_{10}/10\equiv B_{6}/6\pmod5$, the phase‑transition condition $p=3$ for a three‑state Potts model, the product $\zeta(-1)^{-1}(1-p^{k-1})^{-1}=12$ for $p=2$, $k=2$ (while $(1-p^{k-1})^{-1}=-1$), the theta‑function interval $L(2,\vartheta)\in[1.50111,1.51568]$, the root‑of‑unity theta sum $|\sum_{n\ge1}2^{-n}i^{n^{2}}|=\sqrt5/3\approx0.745356$, the tree spectral edge $2\sqrt{p}$ (e.g. $2.8284271$ for $p=2$), and a cophenetic‑distance counterexample $d(4,9)=4\gt \max\{d(4,6),d(6,9)\}=2.

   from Abstract

2. Engineers of quantum error‑correcting hardware, number theorists studying automorphic spectra, and physicists modeling holographic dualities all confront a concrete bottleneck: discrete hierarchical substrates (Bruhat–Tits trees, $p$‑adic lattices, prime‑factor trees) produce observable continuous phenomena (hyperbolic resonance patterns, low‑energy spectra) without a transparent computational bridge.

   from 1. Introduction

3. Derivation of explicit arithmetic illustrations: $B_{2}=1/6$, $B_{10}=5/66$, $|3^{2}|_{3}=1/9$, Kummer congruence $B_{10}/10\equiv B_{6}/6\pmod5$, phase‑transition condition $p=3$, reciprocal product $12$, theta‑value interval $[1.50111,1.51568]$, root‑of‑unity theta sum magnitude $\sqrt5/3$, tree spectral edge $2\sqrt{p}$, and cophenetic‑distance counterexample (Section 4).

   from 1. Introduction

Where the claim stops

1. The conjecture rests on several assumptions.

   from 6. Discussion

2. First, we assume that the embedding $\overline{\mathbb Q}\hookrightarrow\mathbb C_{p}$ preserves the zero set of $L(s,\vartheta)$ without introducing spurious $p$‑adic poles.

   from 6. Discussion

3. The conjecture would be falsified if a numerical experiment revealed a zero $\rho$ of $L(s,\vartheta)$ whose $p$‑adic norm $|\rho|_{p}$ does not coincide with any eigenvalue of the tree Laplacian on $\mathcal T_{p}$, or if the transformed cross‑ratio spectrum on $\mathbb H^{2}$ failed to contain $\log|\rho|$.

   from 6. Discussion

Where to read

1. Abstract159 words · 1 min
2. 1. Introduction212 words · 1 min
3. 2. Background and Related Work337 words · 1 min
4. 3. Methods252 words · 1 min
5. 4. Analysis403 words · 2 min
6. 5. Results124 words · 1 min
7. 6. Discussion230 words · 1 min
8. 7. Practical Implications and Engineering Pathways320 words · 1 min
9. 8. Conclusion100 words · 1 min
10. Correction Notice380 words · 2 min
11. Provenance and AI Involvement242 words · 1 min
12. Appendix A. Divergence report21 words · 1 min