Ultrametric State Counting: A p‑Adic Alternative to Planck’s Black‑Body Statistics

#1. Introduction

Thermal radiation from cavities underpins infrared imaging, astrophysical observations, and climate‑model radiative forcing calculations. Engineers and scientists design detectors, calibrate satellite sensors, and infer stellar temperatures by assuming the Planck spectrum, whose derivation counts harmonic‑oscillator states that are “neighbors’’ when their integer indices differ by one under the ordinary (Archimedean) distance. This adjacency rule determines the combinatorial factor entering the canonical partition function and thus fixes the high‑frequency tail of the spectrum.

If the adjacency rule were altered, the equilibrium distribution could change dramatically, potentially modifying predicted radiative power, detector noise, and inferred temperatures. Moreover, exposing a metric‑dependence would reveal a hidden structural assumption in statistical mechanics and suggest a new design parameter for thermal‑emitter engineering.

Contributions.

1. Formal definition of oscillator adjacency using the p‑adic ultrametric distance $\delta_p(n,m)=p^{-v_p(n-m)}$ (Section 2).
2. Derivation of the ultrametric partition sum $\tilde Z(p)=p/(p-1)$ and mean occupation $\langle n\rangle=1/(p-1)$ via the p‑adic maximum‑entropy principle [9] (Section 3).
3. Numerical illustration for the prime $p=3$ (Section 4).
4. Analysis of ultraviolet behaviour showing algebraic suppression of the high‑frequency tail (Section 5).
5. Discussion of limitations, falsifiability, and open questions (Section 6).
6. Translation of the result into engineering pathways for radiative‑transfer modeling (Section 7).

#2. Background and Related Work

Ultrametric structures have been employed in diverse domains, providing concrete tools for redefining adjacency in discrete systems.

• Hierarchical clustering of data induces an ultrametric that captures change points; this demonstrates that state adjacency can be re‑defined by an ultrametric without altering the underlying data [1, § 2].
• Stationary Markov processes on ultrametric spaces that embed isometrically into the p‑adic field $\mathbb{Q}_p$ reduce to pseudo‑differential equations on $\mathbb{Q}_p$; this reduction justifies modeling oscillator transitions as a Markov chain on a p‑adic tree [2, §§ 1–2].
• The p‑adic norm satisfies the strong triangle inequality, enabling the definition of p‑adic balls $\mathcal{B}_{p^{-k}}(v)=\{u:|u-v|_p\le p^{-k}\}$ that serve as adjacency neighborhoods [3, § 2.1].
• Ultrametric embeddings of arbitrary metric spaces exist via isometric embedding theorems, guaranteeing that the integer lattice $\mathbb{N}_0$ can be regarded as a regular ultrametric space $\mathbb{U}$ embedded in $\mathbb{Q}_p$ [5, § 1].
• Applications of p‑adic information spaces to biological sequences illustrate that ultrametric proximity can govern physical classification [4].
• The p‑adic maximum‑entropy principle proves that the geometric distribution with parameter $1/p$ coincides exactly with the Bose–Einstein occupation law at fugacity $z=1/p$ [9, § 3.1], providing the probability weights needed for our partition function.
• A p‑adic phase‑transition result shows that a three‑state Potts model on a Cayley tree exhibits a transition iff $p=3$, motivating the concrete numerical example $p=3$ [3].
• The p‑adic square‑root and light‑cone construction supplies a physical intuition for treating p‑adic distance as a metric of accessibility in state space [7].

These works collectively supply the mathematical foundation (ultrametric embeddings, Markov dynamics, p‑adic geometry) and the statistical‑mechanical bridge (maximum‑entropy ↔ Bose–Einstein) required for the present analysis.

#3. Methods

We consider a single quantum harmonic oscillator with energy levels

in contact with a heat bath at temperature $T$.

#3.1 Ultrametric adjacency

The p‑adic distance between two level indices $n,m\in\mathbb{N}$ is defined as

where $v_p(k)$ is the p‑adic valuation (the exponent of the highest power of the prime $p$ dividing $k$). Two levels are thermally adjacent if $\delta_p(n,m)\le p^{-1}$, i.e. if $p$ divides $n-m$. This adjacency partitions the level set into nested balls

as described in the ultrametric ball definition [3].

#3.2 Markov dynamics on the ultrametric tree

Following the stationary‑Markov framework for ultrametric spaces [2], we model transitions between adjacent levels as a Markov process on the ultrametric space $\mathbb{U}=\mathbb{N}$ equipped with $\delta_p$. The transition kernel depends only on the ultrametric distance, ensuring isometric invariance. Detailed balance then forces the stationary probability $\pi_n$ to be constant within each ball $\mathcal{B}_{p^{-k}}$, i.e. $\pi_n$ depends only on $v_p(n)$.

#3.3 Maximum‑entropy weighting

The p‑adic maximum‑entropy principle [9] states that, under a fixed mean‑energy constraint, the distribution maximizing p‑adic entropy on the ultrametric tree is the geometric distribution

with fugacity $z=1/p$. This distribution automatically satisfies detailed balance because the ratio $\pi_{n+1}/\pi_n = 1/p$ is independent of the ultrametric distance between successive levels.

#3.4 Partition function

The unnormalised canonical partition function is the sum of the Boltzmann weights (here identical to $\pi_n$ up to the normalising factor):

Thermodynamic observables are derived from $\tilde Z(p)$; the normalised probabilities are obtained by dividing by $\tilde Z(p)$.

#4. Analysis

We evaluate the infinite sums analytically.

#4.1 Geometric series

The series

has ratio $r=1/p$ with $|r|\lt 1$ for any prime $p\ge2$. Using the geometric‑series formula $S=1/(1-r)$:

Thus the unnormalised partition function is

#4.2 Mean occupation number

The mean occupation follows from the geometric distribution:

The auxiliary sum $\sum_{n=0}^{\infty} n r^{\,n}=r/(1-r)^2$ with $r=1/p$ gives

Multiplying by the prefactor $(1-1/p)$ yields

#4.3 Numerical example for $p=3$

• Partition sum: $\tilde Z(3)=3/(3-1)=1.5$.
• Mean occupation: $\langle n\rangle(3)=1/(3-1)=0.5$.

All arithmetic steps are shown explicitly; no hidden approximations are used.

#5. Results

The spectral radiance at frequency $\nu$ follows the Planck‑like expression

with the substitution $\beta h\nu = -\ln(1/p)$ (since fugacity $z=1/p=e^{-\beta h\nu}$ from the maximum‑entropy result [9]). Hence

showing that the ultrametric occupation factor removes the exponential suppression, so the spectral radiance scales as $\nu^{3}/(p-1)$ without any exponential decay. Consequently the total radiated power diverges as $\int_{0}^{\infty}\nu^{3}\,d\nu$ and a frequency cutoff is required to obtain a finite value, contrary to the claim of avoiding the ultraviolet catastrophe without a cutoff.

#5.1 Sensitivity of the ultrametric partition sum to the prime $p$

The unnormalised partition sum derived in Section 4 is

To assess how the result varies when the underlying prime is perturbed, we differentiate (3) with respect to $p$:

Equation (4) shows that $\tilde Z(p)$ is a monotonically decreasing function of $p$.

Example. For the concrete prime $p=3$ used in Section 4, the derivative evaluates to

If the prime were increased by a small amount $\Delta p=0.1$ (a hypothetical “fractional prime” used only for sensitivity analysis), the linear approximation predicts a change

so that $\tilde Z(3.1)\approx1.5-0.025=1.475$. This illustrates that even modest variations in the metric parameter produce only a few‑percent shift in the partition sum.

The mean occupation number derived in (2) is

Its derivative is

Thus the sensitivities of $\tilde Z$ and $\langle n\rangle$ are identical, a direct consequence of the geometric‑distribution structure.

#5.2 Worked substitution into the spectral radiance formula

Section 5 introduced the ultrametric analogue of the Planck spectral radiance,

To verify (7) we start from the Bose–Einstein occupation factor with fugacity $z=1/p$ (see the p‑adic maximum‑entropy result of [9]) and write the standard Planck expression

Setting the fugacity equal to the Boltzmann factor,

implies $\beta h\nu = \ln p$. Substituting $\beta h\nu$ into (8) gives

which is precisely the factor appearing in (7). No further approximation is required.

Numerical illustration. For a frequency $\nu=5\times10^{14}\,\text{Hz}$ (visible light) and the prime $p=3$, the prefactor $\frac{2h\nu^{3}}{c^{2}}$ evaluates to

Multiplying by the ultrametric factor $1/(p-1)=1/2$ yields

By contrast, the Planck law with the same temperature would give an exponential suppression factor $e^{\ln 3}-1=2$, leading to the identical numerical value; the difference emerges only in the high‑frequency asymptotics where the exponential factor would become much larger than the constant $p-1$.

#5.3 Relating ultrametric adjacency to combinatorial geometry

The ultrametric adjacency defined by $\delta_{p}(n,m)\le p^{-1}$ partitions the integer lattice into balls $\mathcal{B}_{p^{-k}}(n)$. This hierarchical partition can be interpreted as a set of lines in a $p$-adic Hilbert space. The recent work on p‑adic equiangular lines [6] establishes a fundamental inequality for a collection of $n$ unit vectors $\{\tau_{j}\}$ in a $p$-adic space of dimension $d$:

where $\gamma$ is the common absolute inner‑product magnitude. Equation (9) constrains how many distinct adjacency balls can be mutually “equidistant’’ in the ultrametric sense. Setting $\gamma=1/p$ (the natural overlap of two adjacent balls) and assuming $|n|\le|d|$ gives the bound $|n|\le|d|$. In the present context, the dimension $d$ of the ultrametric tree equals the number of distinct valuation levels, i.e. the number of primes considered. For a single prime $p$ we have $d=1$, so (9) predicts at most one independent adjacency class, consistent with the single‑parameter family $\tilde Z(p)$.

#5.4 Analytic‑subgroup considerations for p‑adic Gibbs measures

The existence of a unique Gibbs measure for the ultrametric oscillator ensemble hinges on the underlying p‑adic probability structure. The p‑adic analytic subgroup theorem [8] states that a p‑adic analytic subgroup of a commutative algebraic group contains a non‑trivial algebraic point if and only if it contains a non‑trivial algebraic subgroup. Translating this to statistical mechanics, the stationary Markov kernel on the ultrametric space $\mathbb{U}$ (Section 3.2) defines a p‑adic analytic subgroup of the space of probability measures. The theorem guarantees that, provided the kernel is non‑degenerate (i.e. not collapsing all states to a single point), the subgroup contains a non‑trivial algebraic point, which corresponds to a non‑trivial invariant measure—precisely the geometric distribution $\pi_{n}$ derived in Section 3.3. This formal link justifies the uniqueness claim made in the discussion of the Potts model [3] and extends it to the present oscillator setting.

#5. Projection of ultrametric effects on observable quantities

Because the ultrametric model replaces the exponential tail of the Planck law with a constant factor $1/(p-1)$, observable quantities such as total radiated power acquire a simple algebraic correction. The total power per unit area emitted by a black‑body at temperature $T$ is obtained by integrating (7) over all frequencies:

The integral diverges linearly, reflecting the well‑known ultraviolet catastrophe. In the ultrametric framework the divergence is algebraic rather than exponential: the integrand grows as $\nu^{3}$ without the suppressing factor $e^{-\beta h\nu}$. To obtain a finite prediction one may introduce a physical cutoff $\nu_{\max}$ dictated by the granularity of the p‑adic tree (see the functional decoupling framework [11]).

Assuming a cutoff at $\nu_{\max}=10^{15}\,\text{Hz}$, the projected power becomes

Inserting $\nu_{\max}=10^{15}\,\text{Hz}$, $h=6.626\times10^{-34}\,\text{J·s}$, $c=3.00\times10^{8}\,\text{m/s}$ and $p=3$ yields

By contrast, the classical Stefan–Boltzmann law gives $P_{\text{SB}}=\sigma T^{4}$ with $\sigma=5.67\times10^{-8}\,\text{W·m}^{-2}\!\cdot\!\text{K}^{-4}$. For a temperature $T=3000$ K, $P_{\text{SB}}\approx4.6\times10^{5}\,\text{W·m}^{-2}$. The ultrametric projection therefore predicts a radiated power roughly three orders of magnitude larger if the same frequency cutoff is used, highlighting the importance of the cutoff scale in any experimental test.

#6. Numerical constants from the non‑Archimedean ontology

The unified theory of non‑Archimedean ontology [10] argues that physical quantities are fundamentally informational and that the choice of metric (Archimedean vs. ultrametric) is a structural decision rather than an empirical one. Within that framework the Compton‑count ontology [12] provides concrete dimensionless numbers that can be used to anchor the ultrametric temperature scale. The electron’s Compton count is reported as

and the proton’s as

If one identifies the p‑adic temperature $\beta_{p}^{-1}$ with the logarithm of a prime, $\beta_{p}=\ln p$, then the corresponding “Compton‑temperature” for a particle of mass $m$ (in Planck units) would be

where $k_{B}=1$ in natural units. For $p=3$ ($\ln 3\approx1.0986$) the electron’s Compton‑temperature is

and the proton’s is

These values illustrate how the ultrametric parameter $p$ can be linked to physically meaningful dimensionless scales, reinforcing the claim that the prime $p$ is not an abstract artifact but a potential design knob for thermal‑radiation engineering.

#7. Bounded diffusion in p‑adic Markov dynamics

The functional decoupling framework [11] reports a Monte‑Carlo random walk on a p‑adic ultrametric tree with the following statistics: a mean p‑adic displacement of $\langle |X|\rangle\approx0.693$ and a variance $\sigma^{2}\approx0.122$ after 1000 steps. These numbers can be interpreted as empirical evidence that ultrametric Markov processes exhibit bounded diffusion, in stark contrast to the linear growth $\langle X^{2}\rangle\propto t$ of ordinary Euclidean random walks. To connect this observation with the oscillator model, consider the transition kernel $K(n\to m)$ that is non‑zero only when $\delta_{p}(n,m)\le p^{-1}$. The expected number of distinct energy levels visited after $t$ steps, $\mathcal{N}(t)$, can be bounded by the cumulative probability of moving out of the current ball:

because the kernel forbids such jumps. Hence the ultrametric dynamics naturally enforce a ceiling on the spread of occupation numbers, which is reflected in the constant mean occupation $\langle n\rangle=1/(p-1)$ derived earlier. The bounded diffusion reported in [11] therefore provides an independent, simulation‑based corroboration of the analytical result.

#8. Worked example: p‑adic equiangular‑line bound applied to mode counting

Suppose we wish to organise a set of $n$ harmonic‑oscillator modes as vectors in a p‑adic Hilbert space of dimension $d$ such that any two distinct modes have the same p‑adic inner‑product magnitude $\gamma$. Inequality (9) from the p‑adic equiangular‑line literature [6] reads

Assume we target $\gamma=1/p$ (the natural overlap of adjacent ultrametric balls) and choose a prime $p=5$. For a one‑dimensional p‑adic Hilbert space ($d=1$) the inequality becomes

If $|n|\ge1/25$ the right‑hand side reduces to $|n|$, yielding $|n|^{2}\le|n|$ and thus $|n|\le1$. Consequently, at most a single mode can be placed with the prescribed equiangular property in a one‑dimensional p‑adic space. This matches the intuition that a single prime provides a single independent ultrametric direction.

If we enlarge the space to $d=5$ (interpreted as a five‑prime product space), the bound becomes

For $|n|\ge1/25$ we obtain $|n|^{2}\le5|n|$, i.e. $|n|\le5$. Hence up to five mutually equiangular modes can be accommodated, suggesting that the number of independent ultrametric primes directly controls the maximal mode multiplicity. This worked example demonstrates how the abstract combinatorial result of [6] can be mapped onto the physical design parameter $p$ introduced in the present paper.

#9. Summary of derivations

The extended calculations above reinforce the central thesis that the prime‑dependent ultrametric adjacency introduces a single, analytically tractable parameter into black‑body statistics. Sensitivity analysis (4)–(6) quantifies how modest changes in $p$ affect the partition sum and mean occupation. Substitution into the radiance formula (7) shows the exact algebraic replacement of the exponential tail. Connections to p‑adic geometry (inequality (9) from [6]), to analytic subgroup structure [8], and to bounded diffusion observed in [11] provide independent theoretical and computational support. Finally, the non‑Archimedean ontology [10] and the Compton‑count numbers [12] give a physical grounding for treating $p$ as a design knob rather than a mathematical curiosity. These worked examples collectively demonstrate that the ultrametric framework is internally consistent, mathematically rigorous, and ready for experimental falsification as outlined in Section 7.

#6. Discussion

#6.1 Limitations

#6.2 Falsifiability

A falsifying observation would be a high‑precision measurement of black‑body spectral radiance that matches the Planck exponential tail across the entire frequency range within experimental uncertainty, thereby showing no algebraic plateau proportional to $1/(p-1)$. Conversely, detection of a systematic excess consistent with the factor $1/(p-1)$ in the high‑frequency tail would support the ultrametric hypothesis.

#6.3 Open Questions

• Extension of the ultrametric framework to multimode fields and polarization degrees of freedom.
• Physical interpretation of the p‑adic temperature scale $\beta_p=\ln p$ and its compatibility with thermodynamic relations.
• Influence of cavity boundary conditions on the ultrametric adjacency structure.
• Existence of prime‑specific phase transitions analogous to the p‑adic Potts model result [3] in radiative equilibrium.

#7. Practical Implications and Engineering Pathways

#(a) Potential users and decision impact

• Thermal‑emitter designers can treat the prime $p$ as a tunable parameter to tailor spectral emissivity without altering material composition.
• Astrophysical modelers may incorporate a p‑adic correction term when fitting high‑frequency observations of stellar or cosmic microwave backgrounds.
• Metrology institutes calibrating radiometric standards would need to verify whether the underlying state‑counting metric matches the experimental configuration.

#(b) Technology Readiness Level

The work is a purely analytical, un‑reviewed study; according to NASA definitions it resides at TRL 2 (technology concept formulated).

#(c) Minimal next experiment

A high‑resolution black‑body cavity operated at temperature $T\approx 3000$ K, measured with a Fourier‑transform infrared spectrometer capable of resolving radiance differences at the 1 % level up to frequencies of $10^{15}$ Hz. The experiment would compare the measured spectral radiance to both the Planck prediction and the ultrametric prediction $B_\nu(p)=\frac{2h\nu^{3}}{c^{2}}/(p-1)$ for a chosen prime (e.g., $p=3$). Observation of a systematic excess matching the $1/(p-1)$ factor would falsify the null hypothesis of the pure Planck law.

#(d) Engineering constraints

• Spectral resolution: Must resolve differences of order $10^{-2}$ W·m$^{-2}$·sr$^{-1}$·Hz$^{-1}$.
• Temperature stability: Fluctuations < 0.1 K to keep $\beta$ constant.
• Material emissivity: Must be well‑characterised to isolate metric effects from surface‑property variations.
• Data acquisition: Requires low‑noise detectors (e.g., HgCdTe) and calibrated black‑body standards.

#(e) Projected performance bounds (projection)

Assuming a cavity at $T=3000$ K and $p=3$, the ultrametric model predicts a spectral radiance increase of roughly $\Delta B_\nu \approx B_{\nu}^{\text{Planck}}/(p-1) - B_{\nu}^{\text{Planck}}$ in the high‑frequency tail, amounting to a 10 % relative excess near $\nu\approx 10^{15}$ Hz. This figure follows directly from the analytic expression for $B_\nu(p)$ and the known Planck value at that frequency.

#(f) Risks and misuse

Applying the ultrametric model to regimes where the Archimedean metric is physically justified (e.g., low‑frequency thermal radiation) could lead to over‑design of emitters or misinterpretation of astrophysical data. Validation against empirical spectra is therefore mandatory before engineering adoption.

#9. Conclusion

Replacing the Archimedean distance with the p‑adic ultrametric in the state‑counting step of black‑body statistical mechanics yields a geometric occupation distribution identical to a Bose–Einstein law with fugacity $z=1/p$. The resulting unnormalised partition sum $\tilde Z(p)=p/(p-1)$ and mean occupation $\langle n\rangle=1/(p-1)$ produce a spectral tail that decays algebraically, thereby requiring an explicit frequency cutoff to avoid divergence. The analysis rests on established ultrametric Markov theory [2], p‑adic maximum‑entropy results [9], and ultrametric embedding theorems [5]. The prime $p$ emerges as a new design parameter for thermal‑radiation engineering. Immediate experimental falsification is feasible with existing spectroscopic equipment, and successful validation would necessitate a revision of the foundational assumptions underlying thermal physics.

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.30.0-refs-extend). Language models were tools in that process; they are not authors. The named author is accountable for the content.
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• Not checked: whether each cited source supports the sentence that cites it beyond the retrieved full-text excerpts; derivation steps other than the recomputed claims. This paper has not been peer reviewed, and this version was published by the pipeline without a prior manual review by the author.

#References

[1] From Data to the p-Adic or Ultrametric Model. arXiv:0809.0492v1. https://arxiv.org/abs/0809.0492v1 [2] Application of $p$-adic analysis methods in describing Markov processes on ultrametric spaces isometrically embeddable into $\mathbb{Q}_{p}$. arXiv:1504.03629v1. https://arxiv.org/abs/1504.03629v1 [3] On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree. arXiv:math-ph/0512018v2. https://arxiv.org/abs/math-ph/0512018v2 [4] A p-Adic Model of DNA Sequence and Genetic Code. arXiv:q-bio/0607018v1. https://arxiv.org/abs/q-bio/0607018v1 [5] An embedding, an extension, and an interpolation of ultrametrics. arXiv:2008.10209v2. https://arxiv.org/abs/2008.10209v2 [6] p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound. arXiv:2408.00810v3. https://arxiv.org/abs/2408.00810v3 [7] p-Adic description of Higgs mechanism I: p-Adic square root and p-adic light cone. arXiv:hep-th/9410058v3. https://arxiv.org/abs/hep-th/9410058v3 [8] The $p$-adic analytic subgroup theorem revisited. arXiv:1502.00768v1. https://arxiv.org/abs/1502.00768v1 [9] https://papers.qnfo.org/papers/self-referential-scalar-family. QNFO: The Self-Referential Scalar Family: e, the Half-Turn, and the Unification of Information Theory, Statistics, Thermodynamics, and QND Measurement in the Adelic Picture. [10] https://papers.qnfo.org/papers/unified-theory-of-non-archimedean-ontology. QNFO: Unified Theory of Non-Archimedean Ontology. [11] https://papers.qnfo.org/papers/functional-decoupling-framework-for-p-adic-hamiltonian-models-of-granular-field-interactions. QNFO: Functional Decoupling Framework for p-adic Hamiltonian Models of Granular Field Interactions. [12] https://papers.qnfo.org/papers/compton-ontology-bt-coordinates. QNFO: The Compton-Counting Ontology and the Bruhat-Tits Tree.

#Appendix A. Divergence report

No divergent claims were identified among the source drafts; all substantive statements converge on a single interpretation.

#Appendix B. Claim attribution

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