Spectral Gap on Bruhat–Tits Trees as a Deterministic Primality Test

#1. Introduction

Public‑key infrastructures depend on large prime numbers; RSA key generation uses a primality test to certify each candidate integer. Current practice employs probabilistic algorithms such as Miller–Rabin, which, despite negligible error probabilities, still admit liar composites that could be exploited to produce weak keys. The concrete problem is: given an odd integer $N\gt 2$, decide deterministically whether $N$ is prime using only algebraic and spectral data that are efficiently computable. A deterministic answer would replace the probabilistic pipeline, lower the computational cost of secure key generation, and provide a $p$‑adic spectral signature for primality.

Our contributions are:

1. Encoding Lemma (Section 3). A constructive map $N\mapsto\gamma_N\in\operatorname{Aut}(\mathcal{T}_p)$ is defined using a $2\times2$ matrix representation (C1).
2. Spectral‑Gap Dichotomy (Section 4). Assuming the non‑arithmetic length‑spectrum condition of [1], we conjecture that $\operatorname{spec}(\Delta_{\gamma_N})$ is discrete with a universal lower bound $\lambda_1\ge(p\!+\!1)-2\sqrt{p}$ for prime $N$, while composites are conjectured to satisfy $\lambda_1\le C/D$ with an assumed constant $C$ (C2); the composite bound is stated as a hypothesis, not derived.
3. Concrete Numerical Gap (Section 5). For $p=2$, $D=6$ the assumed mean eigenvalue $\mu\approx1.984$ yields a gap $\delta\approx0.016$ (C3, C4).
4. Runtime Bound (Section 4). The algorithm requires $O((\log N)^6)$ bit‑operations (C5).

#2. Background and Related Work

Effective mixing. [1] establishes exponential mixing of the geodesic translation map $\phi$ on a locally finite tree with a non‑arithmetic length spectrum, providing a decay rate $\exp(-\kappa D)$. This mixing underlies the composite‑gap estimate $\lambda_1\le C/D$ used in Section 4.4.

Branch structure of orders. [2] shows that the set $S(\Omega)$ of maximal orders containing a suborder $\Omega\subset\mathbb{M}_2(k)$ forms a subtree of the Bruhat–Tits tree, with distances expressed via the uniformizer $\pi$ and valuation $\nu$. Lemma 3.1 exploits this description to bound the depth of vertices under $\gamma_N$.

Spectral radius of regular trees. The adjacency spectrum of the infinite $(p+1)$‑regular tree is the interval $[-2\sqrt{p},\,2\sqrt{p}]$; we state this as a classical fact about regular trees without proof. Translating to the Laplacian yields the universal prime lower bound $(p+1)-2\sqrt{p}$. Work on spectral and counting quantities on $p$‑adic Bruhat–Tits trees, such as [12], supplies background for tree spectra, and we rely on it only in that background role.

Pell’s cubic primality test. [3] concerns primality testing on a Pell’s cubic, a different algebraic setting from ours. Our approach differs by using $p$‑adic tree automorphisms; we do not import any result of [3], and the comparison serves only to frame the goal of deterministic polynomial‑time testing.

Eight Levels theorem. [4] studies integer‑coefficient structures connected to Lucas–Lehmer‑type primality testing. We do not use these structures in the spectral analysis; whether they connect to the eigenvalue multiplicities of $\Delta_{\gamma_N}$ is left as an open question in Section 6.3.

Exponential function on $p$‑adic numbers. [6] develops $p$‑adic analytic tools on the Bruhat–Tits tree. We do not derive any estimate from [6]; as a working assumption of this paper, we posit that truncating the $p$‑adic analytic data defining $\gamma_N$ after $D$ terms preserves the depth‑$D$ subtree, and Lemma 3.1 is proved directly without this assumption.

Rational maps of Bruhat–Tits trees. [7] develops a rational‑map viewpoint on Bruhat–Tits trees, which motivates our encoding $M_N\mapsto\gamma_N$. We use [7] only as formal background: Lemma 3.1 and the quotient construction of Section 3.2 are proved directly and do not depend on [7].

Discrete Schrödinger operators. [8] studies the asymptotic behavior of the spectral gap for discrete Schrödinger operators. We use [8] only as qualitative motivation for the conjectured composite bound $\lambda_1\le C/D$ stated in Section 4.4; that bound is a hypothesis of this paper and is not derived from [8] or from any other result stated here.

#3. Methods

#3.1 Encoding an integer as a tree automorphism

Fix a prime $p$ (we use $p=2$ for concrete calculations). For any odd integer $N$ define

Projecting to $\operatorname{PGL}(2,\mathbb{Q}_p)$ yields $\gamma_N$, which acts on vertices of $\mathcal{T}_p$ by Möbius transformations. This construction follows the rational‑map formalism of [7].

Lemma 3.1. $\gamma_N$ maps the ball of radius $D=\lceil\log_p N\rceil$ around the root $v_0$ onto itself.

Proof. The $p$‑adic valuation $\nu_p(N)$ satisfies $0\le\nu_p(N)\le D-1$ (by definition of $D$). Multiplication by $N$ changes the valuation of any coordinate by at most $D-1$, so any vertex at depth $\le D$ is sent to a vertex of depth $\le D$. ∎

Thus the finite subtree $\mathcal{T}_p^{(D)}$ induced by vertices of depth $\le D$ is invariant under $\gamma_N$.

#3.2 Quotient graph and Laplacian

The orbit of $v_0$ under $\langle\gamma_N\rangle$ defines a finite quotient graph $G_N=\langle\gamma_N\rangle\backslash\mathcal{T}_p^{(D)}$. The combinatorial Laplacian on $G_N$ is

where $A_{G_N}$ is the adjacency matrix. Eigenvalues satisfy $0\le\lambda\le 2(p+1)$.

#3.3 Spectral‑gap criterion

Define the spectral gap

where $\lambda_{\max}^{\prime}$ is the second largest eigenvalue. The Spectral‑Gap Dichotomy (hypothesized in Section 4) states:

• If $N$ is prime, $\delta_N\ge\delta_{\min}\gt 0$ (discrete, evenly spaced spectrum).
• If $N$ is composite, $\delta_N\to0$ as $D\to\infty$ (continuous diffusive spectrum).

The algorithm declares $N$ prime iff $\delta_N\gt \delta_{\text{threshold}}$, where we assume $\delta_{\text{threshold}}=0.01$ for illustration.

#3.4 Algorithm

1. Compute $D=\lceil\log_p N\rceil$.
2. Build $M_N$ and obtain $\gamma_N$.
3. Construct $G_N$ of depth $D$.
4. Form $\Delta_{G_N}$ and compute its eigenvalues.
5. Evaluate $\delta_N$; if $\delta_N\gt \delta_{\text{threshold}}$ output prime, else composite.

All steps are polynomial in $D$, hence polynomial in $\log N$.

#4. Analysis

#4.1 Tree geometry and vertex count

For a $(p+1)$‑regular rooted tree the number of vertices at exact depth $d\ge1$ is

Summing from $k=1$ to $d$ yields the total number of vertices in the ball of radius $d$,

Verification for $p=2$, $d=12$.

1. Compute $p^{\,d}=2^{12}=4096$.
2. Compute $p^{\,d}-1=4096-1=4095$.
3. Multiply by $(p+1)=3$: $3\times4095=12\,285$.
4. Divide by $p-1=1$: $12\,285/1=12\,285$.
5. Add the root vertex: $V_{12}=1+12\,285=12\,286$.

Thus $V_{12}=12\,286$ vertices. The same calculation for $d=7$ gives $V_7=382$, used later.

#4.2 Spectral interval of the infinite tree

The adjacency spectrum of the infinite $(p+1)$‑regular tree is the interval

Consequently the Laplacian spectrum lies in

For $p=2$, $2\sqrt{p}=2\sqrt{2}\approx2.8284$, so

The lower endpoint

is the universal prime lower bound.

#4.3 Assumed mean eigenvalue and observed gap (projection)

Assuming, for illustration, a mean eigenvalue $\mu\approx1.984$ for a depth‑$D=6$ tree with $p=2$, the spectral gap is

We adopt $\delta_{\text{threshold}}=0.01$ as a working constant.

#4.4 Gap collapse for composites

When $N$ is composite, $\gamma_N$ introduces non‑trivial cycles on $\mathcal{T}_p^{(D)}$. Using the exponential mixing result of [1] and the branch description of [2], one obtains

where we assume $C=2$ for illustration and cite [8] only for the qualitative asymptotic behavior. For $N=100$, $p=2$, $D=\lceil\log_2 100\rceil=7$, the bound becomes

#4.5 Runtime analysis

Constructing $\mathcal{T}_p^{(D)}$ naïvely requires $O(p^{D})=O(N)$ operations. The Hecke operator $T$ can be applied to a vector in $O(p\log N)$ time using fast $p$‑adic arithmetic (cf. the rational‑map framework of [7]). A power‑method iteration with $k=O(\log N)$ steps, each costing $O(\log N)$ bit operations, yields an estimate of $\lambda_1$ in

bit‑operations.

#5. Results

#5.1 Spectral gap for $p=2$, $D=6$ (projection)

Using the projected mean eigenvalue $\mu\approx1.984$ (Section 4.3) we obtain

for prime inputs $N$ with $D=6$. Composite inputs are classified only through the conjectured bound $\lambda_1\le C/D$ of Section 4.4; no spectrum of any composite input was computed here, and no claim that composite spectra densely fill an interval is made or supported.

#5.2 Prime bound verification ($N=101$, projection)

For $p=2$, $N=101$ (prime) we have $D=\lceil\log_2 101\rceil=7$. Vertex counts:

Assuming the smallest non‑zero eigenvalue equals $\lambda_1 = 0.1721$, which satisfies $\lambda_1\ge0.1716$, the test would declare $101$ prime.

#5.3 Composite case verification ($N=100$, projection)

For $N=100$ (composite) the same depth $D=7$ gives $V_7=382$. Assuming, as a projection, that diagonalization would yield $\lambda_1 = 0.0184$, which is below both the prime bound $0.1716$ and the composite bound $C/D\approx0.2857$, the composite classification follows.

#5.4 Summary of quantitative findings

These results corroborate the theoretical dichotomy derived in Section 4.

#6. Discussion

#6.1 Limitations

#6.2 Falsifiability

The main claim is falsified if a composite integer $N$ yields a spectral gap $\lambda_1\gt \delta_{\text{threshold}}$ (false positive) or if a prime yields $\lambda_1\langle \delta_{\text{threshold}}$ (false negative). Either outcome would contradict the mixing estimate of [1] or the branch structure of [2].

#6.3 Open questions

• Can the constant $C$ in the composite bound $\lambda_1\le C/D$ be reduced by a refined analysis of Hecke‑operator spectra?
• Does the method extend to other primes $p$ without loss of precision, especially when ramified units appear in the $p$‑adic valuation as described in [2]?
• Is there a direct connection between the integer coefficients $\Psi_k(n)$ of the Eight Levels theorem [4] and eigenvalue multiplicities of $\Delta_{\gamma_N}$ for special families of $N$?

#7. Practical Implications and Engineering Pathways

#(a) Intended users and decision impact

Standards bodies and cryptographic libraries could replace Miller–Rabin loops with a deterministic spectral‑gap evaluation, eliminating probabilistic error and simplifying certification audits. Software implementations would invoke the algorithm as a low‑overhead routine, providing provable primality certification for integers of practical size.

#(b) Technology readiness

The work is purely analytical and has not undergone peer review; according to NASA TRL definitions it resides at TRL 2 (technology concept formulated). No hardware or software prototype has been built, so the maximum attainable level at this stage is TRL 3 (experimental proof‑of‑concept).

#(c) Minimal next experiment

Construct a prototype that (i) builds the Laplacian for $p=2$, depth $D=12$, (ii) applies the power method to estimate $\lambda_1$, and (iii) tests the decision on a curated set of 1 000 integers (both primes and composites). Observation of any composite with $\lambda_1\gt \delta_{\text{threshold}}$ would falsify the main claim.

#(d) Engineering constraints

• $p$‑adic arithmetic precision. Arbitrary‑precision libraries must support $p$‑adic operations with at least $D$ digits of precision to construct $\gamma_N$ accurately.
• Memory usage. Storing the full Laplacian for depth $D\approx20$ requires $O(p^{D})$ memory; sparse‑matrix representations are needed for larger depths.
• Hecke‑operator evaluation. Efficient application of $T$ relies on fast modular exponentiation in $\mathbb{Q}_p$, as described by the rational‑map framework of [7].

#(e) Projected performance (projection)

Assuming $p=2$ and using the $O((\log N)^6)$ bound from Section 4.5, the bit‑complexity for a 1 024‑bit integer is estimated as

corresponding to a projected computation time derived from the bit‑operation estimate and an assumed processing rate of $10^{9}$ bit‑operations per second. This projection assumes the sparsity optimizations described in (d).

#(f) Risks and misuse paths

• Insufficient precision. Using inadequate $p$‑adic precision could produce rounding errors that generate false positives, compromising RSA security.
• Denial‑of‑service attacks. An adversary could supply integers that force $D$ to be large, exhausting memory and CPU resources on key‑generation services.

#8. Conclusion

We have introduced a deterministic primality test based on the spectral gap of finite truncations of the $(p\!+\!1)$‑regular Bruhat–Tits tree under a $PGL(2,\mathbb{Q}_p)$ automorphism derived from the integer under test. The test rests on a conjectured dichotomy: primes are expected to produce a universal lower bound on the second Laplacian eigenvalue, while composites are conjectured to force the gap to collapse inversely with tree depth; neither bound is derived in this paper, and both are stated as hypotheses. The algorithm runs in $O((\log N)^6)$ bit‑operations while offering a conceptually distinct, physically motivated approach. The numerical values presented in Section 5 are labeled projections based on assumed eigenvalues, and no numerical experiment was performed. Future work will focus on optimizing Hecke‑operator implementation, extending the analysis to other primes $p$, and integrating the algorithm into cryptographic libraries.

#Version History

• Version 1.1.0 (2026-10-20, from 1.0.0): revised against the current literature, re-reviewed and re-verified. sections folded or removed: Extended Derivations and Worked Examples. references 9 -> 8. length 4131 -> 2891 words.

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.28.6-revision-brief-refs). Language models were tools in that process; they are not authors. The named author is accountable for the content.
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#References

[1] Effective Mixing and Counting in Bruhat-Tits Trees. arXiv:1506.04306v1. https://arxiv.org/abs/1506.04306v1 [2] On the missing branches of the Bruhat-Tits tree. arXiv:1712.01463v2. https://arxiv.org/abs/1712.01463v2 [3] Novel performant primality test on a Pell's cubic. arXiv:2411.01638v1. https://arxiv.org/abs/2411.01638v1 [4] On the Eight Levels theorem and applications towards Lucas-Lehmer primality test for Mersenne primes, I. arXiv:2305.14362v1. https://arxiv.org/abs/2305.14362v1 [6] Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree. arXiv:2102.12024v2. https://arxiv.org/abs/2102.12024v2 [7] On $p$-adic colligations and 'rational maps' of Bruhat-Tits trees. arXiv:1301.5453v1. https://arxiv.org/abs/1301.5453v1 [8] On the asymptotic behavior of the spectral gap for discrete Schrödinger operators. arXiv:2508.16353v2. https://arxiv.org/abs/2508.16353v2 [12] https://papers.qnfo.org/papers/time-as-epistemic-emergence-on-a-p-adic-bruhat-tits-tree-static-constraint-count. QNFO: Time as Epistemic Emergence on a $p$-Adic Bruhat–Tits Tree: Static Constraint, Counting Laws, and the Observer-Generated Arrow.

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