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A p-adic operator spectrum balances exponential energies and degeneracies to reproduce the string's Hardy-Ramanujan entropy scaling, with log-periodic modulations.

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2026-08-03 12:15 UTC pith:NMM4EOH4

load-bearing objection A genuinely new p-adic spectrum with string-like entropy, but the general-conductor eigenvalue formula needs a proof before the main claim is settled. the 3 major comments →

arxiv 2601.03738 v2 pith:NMM4EOH4 submitted 2026-01-07 hep-th

A glimpse into the Ultrametric spectrum

An Huang , Christian B. Jepsen This is my paper
classification hep-th MSC 11S8011P8281T3005C50
keywords p-adic numbersVladimirov derivativeultrametric spectrumHardy-Ramanujan scalingp-adic string theorylog-periodic oscillationstree normal modesNeumann-to-Dirichlet operator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether an ultrametric, p-adic analog of the string can reproduce the entropy-energy relation of the ordinary non-relativistic string, where the microstate count grows as exp(√E). Its central result is that the eigenvalue spectrum of the Vladimirov derivative on the p-adic unit circle has exponentially growing energies and exponentially growing degeneracies, and these two effects balance so that a saddle-point evaluation of the thermal partition function gives a coarse-grained degeneracy of the form E^(−a) exp(c√E). This recovers the string's Hardy-Ramanujan scaling, but modulated by log-periodic oscillations rather than smooth behavior. The authors also show that more naive tree-graph replacements fail to achieve the same scaling, so the p-adic continuum construction is what makes the string-like counting possible.

Core claim

The paper's central discovery is that the Vladimirov derivative acting on the p-adic unit circle U_p has a spectrum consisting of the trivial character with eigenvalue 0, p−2 characters of conductor one with eigenvalue ε(p−1)/p, and, for n>1, (p−1)^2 p^(n−2) characters of conductor n with eigenvalue E_n = ε[(p+1)/p^2 p^n − 2/p]. Promoting each level to a quantum harmonic oscillator and evaluating the partition function at high temperature gives, after dropping log-periodic fluctuations, a coarse-grained degeneracy Ω(E) ∝ E^(−a) exp[(p−1)π√(2E/ε)/(√(3(p+1) log p))] with a = 5/4 − (p−1)^2/[p(p+1) log p], so log Ω ∼ √E. The same eigenfunctions and multiplicities, and the same leading entropy en

What carries the argument

The key objects are the multiplicative characters of the p-adic units, classified by their conductor n (the number of p-adic digits on which the character depends), and the eigenvalue integral λ_π = ∫_{U_p} dz (1−π(z))/|z−1|_p^2 for the Vladimirov derivative, a bi-local derivative operator on p-adic numbers. Because characters depend on only finitely many digits, the integral collapses to a finite sum over residue classes; the paper evaluates these sums explicitly for conductors 1, 2, and 3 and asserts the general-n result. The thermodynamic argument then turns on the infinite product partition function and a saddle-point evaluation in which the exponential spacing p^n of energies and the ex

Load-bearing premise

The load-bearing premise is that the eigenvalue integral for multiplicative characters of arbitrary conductor n collapses to λ=(p+1)p^(n−2)−2/p by the same method as the three cases actually shown (n=1,2,3), since the whole spectrum—and with it the entropy law—stands on that single unproven general-n formula; the coarse-grained envelope additionally rests on a numerically checked but not analytically proven saddle-point formula.

What would settle it

Enumerate all multiplicative characters of conductor n=4 for p=3 (there are (p−1)^2 p^(n−2)=36 of them), evaluate the defining eigenvalue integral directly as a finite sum over residue classes mod p^n, and check whether every eigenvalue equals (p+1)p^(n−2)−2/p = 36 − 2/3; any deviation, or any dependence on the character label j, falsifies the spectrum and hence the thermodynamics.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The ultrametric spectrum yields a concrete infinite tower of microstates with string-like entropy, opening a path toward p-adic string theories that contain more than a single tachyon.
  • The balance condition—energy spacing p^n with degeneracy (p−1)^2 p^(n−2)—explains why the Neumann-tree spectrum, whose levels grow only as p^(n/2), instead produces S ∼ E^(2/3).
  • Log-periodic oscillations in log temperature are small for p=2 but grow with p, so thermodynamic quantities derived from this spectrum carry modulations that must be controlled in any large-p application.
  • The Neumann-to-Dirichlet variant shares the same eigenfunctions and the same √E entropy envelope but has a tachyonic ground state, so subleading thermodynamics differ while the leading entropy law remains robust.
  • Periodic boundary conditions on trees are severely constrained: vertex-transitive identifications exist only for finitely many sizes, so the p-adic continuum spectrum is not a limit of a periodic tree graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the general-conductor eigenvalue formula survives closer scrutiny, the saddle-point mechanism is generic: any spectrum with E_n ∝ p^n and ρ_n ∝ p^n should produce a √E entropy envelope, so the same counting should appear for other p-adic operators or unramified extensions.
  • The log-periodic modulation means the exact microstate count is a rough, oscillating function; a natural next step would be to seek a Rademacher-style convergent expansion for the ultrametric partition function, analogous to the refinement of Hardy-Ramanujan for ordinary partitions.
  • The opposite regime—exponentially decreasing spacings, as in q-deformed strings—is the mirror image of this construction; whether an analytic continuation links the two families is a natural open question suggested by the formulas.
  • A sharper numerical test of the claimed balance would be to compute exact partition multiplicities for p=5 at moderate energies and check both the exponential coefficient and the period log p of the oscillations; the paper's numerical checks are shown for p=3.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies deformations of the non-relativistic string spectrum motivated by p-adic geometry. After showing that normal modes of (p+1)-regular tree graphs with Neumann, Dirichlet, or periodic boundary conditions do not yield Hardy–Ramanujan microstate scaling (with the periodic case obstructed by Moore-graph nonexistence), the authors compute the spectrum of the Vladimirov derivative acting on functions on the p-adic unit circle U_p. They claim (eq. 66) that the eigenvalues are E_n = ε[(p+1)/p^2 p^n − 2/p] with degeneracies ρ(1)=p−2 and ρ(n>0)=(p−1)^2 p^{n−2}. Associating a quantum harmonic oscillator to each eigenvalue, they evaluate the thermal partition function in a high-temperature saddle-point approximation and obtain a coarse-grained degeneracy (eq. 79) whose logarithm scales as √E, i.e. string-like Hardy–Ramanujan behavior, but with log-periodic fluctuations. Appendix B identifies the same eigenfunctions and degeneracies for a Neumann-to-Dirichlet operator on the Bruhat–Tits tree, with eigenvalues that differ only by an overall shift and normalization.

Significance. If the central derivation is correct, the paper gives a concrete p-adic operator spectrum whose microstate count reproduces the non-relativistic string entropy–energy relation, despite exponentially growing energy levels, and it does so without fitting parameters: the thermodynamic constants A, B, C, c_n are computed in closed form, the zero-point cancellation (67) is explicitly checked, and the log-periodic modulation is a consequence of the calculation rather than an input. The exact degeneracy counting in Section IV is independently checked by the character count, and the numerical comparison in Figure 7 supports the envelope formula. The extension of the saddle-point method to exponentially spaced, exponentially degenerate spectra is a useful technical contribution. However, the general-conductor eigenvalue formula on which the whole spectrum rests is asserted rather than proved, and the derivation of the envelope from the saddle point is partly by analogy with the string case; these gaps must be closed before the central claim can be considered established.

major comments (3)
  1. [§IV, eqs. (64)–(66)] The eigenvalue formula λ = (p+1)p^{n−2} − 2/p for characters of conductor n is the single load-bearing input of the paper: it determines the spectrum (66), the zero-point cancellation (67), and all thermodynamic constants in Appendix A. The text works out n = 1, 2, 3 (eqs. 59, 61, 63) and then states that the general sum (64) reduces to (65) “by the same method.” No general-n evaluation of the character sums is supplied, and the typesetting of (64) is partly garbled, making the claimed collapse difficult to verify. The reader is left with a proof gap, not an identified error, but the gap is load-bearing. A direct derivation—or a precise reference to a standard p-adic harmonic-analysis result—is required before the spectrum can be treated as settled.
  2. [§V, eqs. (76)–(79)] The paper states explicitly that E(β0) in (78) could not be inverted analytically, and then obtains the envelope (79) by reusing the string saddle-point formula (19) with modified values of A, B, C after “dropping the log-periodic fluctuations.” This transfer is not justified in detail: it is not shown that the saddle point of the averaged entropy has the same form once the oscillatory terms in (76) are discarded, nor that the prefactor and the exponent in (79) are the correct envelope rather than an ansatz. Figure 7 is suggestive, but a numerical check is not a proof. The central Hardy–Ramanujan claim rests on this envelope, so the averaging procedure needs to be made precise.
  3. [§IV, eq. (67) and §V] The zero-point energy divergence is regulated by analytic continuation in p / geometric summation, and the authors find that the total zero-point contribution vanishes exactly. This is a particular regulator choice; a zeta-function or cutoff regulator would in general give a different (possibly divergent or finite) ground-state shift. Since the partition function (68) starts without any zero-point term, the low-temperature and subleading thermodynamics of the model depend on this convention. The authors should discuss the physical status of this regulator and, if possible, show that the leading √E scaling is independent of it.
minor comments (5)
  1. [§IV, eq. (64)] The displayed formula for the general-conductor sum is difficult to parse: the exponents, denominators, and summation limits are not typeset cleanly. Please rewrite it with clearly grouped factors and define all summation ranges.
  2. [§I, eq. (3)] The sample spectra E^{(2)}, E^{(3)}, E^{(5)} are presented without stating the chosen normalization of ε. For example, the p=3 values correspond to ε=3/2. Please state the normalization explicitly.
  3. [Appendix A, eq. (A19)] The text says A, B, C, c_n, etc. are “p-independent constants,” but A, B, and C as given in (A20)–(A22) depend explicitly on p. This should be corrected to “p-dependent constants” or similar.
  4. [Figure 7] The three columns of plots are described only in the caption as differing in plot region. Please label the panels and add axis legends, especially for the lower-row plots where the three curves nearly coincide.
  5. [References] Several entries are incomplete or lack journal/volume/page data (e.g. [45], [58], [61]). The arXiv identifiers are present for most items, but the bibliographic details should be completed for publication.

Circularity Check

0 steps flagged

No significant circularity: the Vladimirov spectrum is computed from a direct eigenvalue integral, and the thermodynamic constants are derived rather than fitted; the main weakness is an asserted induction for general conductor n, which is a proof gap rather than a circular reduction.

full rationale

The central derivation is self-contained. The spectrum (66) comes from evaluating the Vladimirov-derivative eigenvalue integral (56) for multiplicative characters; the degeneracies are counts of characters of each conductor. The free-energy expansion (A19) and all constants A, B, C, c_n in (A20)-(A25) are obtained by Poisson resummation and polylogarithm expansions, not by matching to the target entropy. The envelope (79) reuses the general saddle-point formula (19) with these derived constants, so the Hardy-Ramanujan exponent follows from the computed spectrum rather than being an input. Figure 7 compares exact degeneracies with the derived asymptotics, not fitting them. The self-citations ([66], [79], [17]) are pointers to related work and are not load-bearing for the spectrum or thermodynamics. The genuine weakness is that (65) is asserted 'by the same method' after only n=1,2,3 are shown, leaving the full spectrum and all subsequent thermodynamics dependent on an unproved induction; Appendix B's Neumann-to-Dirichlet calculation gives different eigenvalues and does not close this gap. These are proof gaps and correctness risks, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 0 invented entities

The model is economical: one dimensionful scale ε (a unit choice, not fitted), a standard eigenfunction fact (multiplicative characters), and two conventions (operator normalization, zero-point regulator). The genuinely load-bearing unproven input is the general-conductor eigenvalue evaluation (64)→(65). No new physical entities are postulated: the 'ultrametric string spectrum' is a name for the derived operator eigenvalues, and the authors explicitly defer the dynamical completion (oscillator algebra, Virasoro, Lorentzian realization) to future work (Section VI).

free parameters (1)
  • ε (overall energy scale)
    Dimensionful normalization for E_n in (66) — the analogue of ℏω₀. Not fitted to data; cancels from the entropy–energy scaling law.
axioms (6)
  • standard math Multiplicative characters of U_p are eigenfunctions of the modified Vladimirov derivative and form a complete basis
    Section IV, eqs. (53)–(56): Dπ = λ_π π is verified for characters; completeness is standard harmonic analysis on U_p.
  • ad hoc to paper The general-conductor eigenvalue sum (64) evaluates to λ = (p+1)p^{n−2} − 2/p (eq. 65)
    Shown for n = 1, 2, 3 only (eqs. 59, 61, 63; Figs. 5–6); asserted for general n 'by the same method'. The spectrum (66) and Section V rest on it.
  • ad hoc to paper Divergent zero-point energy is regulated by analytic continuation/geometric summation in p, making Σ ρ(n)E_n = 0 (eq. 67)
    Section V; 'arguably the most natural regulator'. A different regulator shifts the ground state (the NtD case (B6) yields a tachyonic shift) but not the asymptotic entropy.
  • domain assumption The coarse-grained envelope, obtained by dropping the log-periodic Fourier modes of the free energy, is the right object for the claim that Hardy–Ramanujan scaling is 'realized'
    Section V: the paper admits the exact degeneracy admits no strict √E law; the claim is thereby by construction about the envelope. Numerical support in Fig. 7.
  • standard math Poisson resummation and polylogarithm asymptotics used in Appendix A are valid on the relevant integration/expansion domains
    Appendix A, eqs. (A8)–(A11); standard analytic tools; the branch-cut cancellation (A16)–(A17) is intricate but tracked explicitly.
  • standard math Singleton's classification of Moore graphs bounds periodic-tree constructions
    Section III C, cited as Singleton (1966); used to conclude no large-L periodic trees exist for p>1.

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read the original abstract

The non-relativistic string spectrum is built from integer-spaced energy quanta in such a way that the high-temperature asymptotics, via the Hardy-Ramanujan formula for integer partitions, reduces to standard two-dimensional thermodynamics. Here we explore deformed realizations of this behavior motivated by $p$-adic string theory and Lorentzian versions thereof with a non-trivial spectrum. We study the microstate scaling that results on associating quantum harmonic oscillators to the normal modes of tree-graphs rather than string graphs and observe that Hardy-Ramanujan scaling is not realized. But by computing the eigenvalues of the derivative operator on the $p$-adic circle and by determining the eigenspectrum of the Neumann-to-Dirichlet operator, we uncover a spectrum of exponentially growing energies but with exponentially growing degeneracies balanced in such a way that Hardy-Ramanujan scaling is realized, but modulated with log-periodic fluctuations.

Figures

Figures reproduced from arXiv: 2601.03738 by An Huang, Christian B. Jepsen.

Figure 1
Figure 1. Figure 1: FIG. 1. The fractal tree for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plot of the values of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of the values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. By identifying boundary vertices of a finite tree, one [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Computation of the eigenvalue for the multiplicative character of conductor two for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Computation of the eigenvalue for the multiplicative character of conductor three for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of the exact ultrametric degeneracies along with the averaged and unaveraged asymptotic values for [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Harmonic extension [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

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Forward citations

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