Ultrametric Marginality: A p-Adic Laboratory for Hierarchical Criticality

 A technical outline of an ongoing research program

1. Motivation: exponent identities from marginal hierarchical systems

In many critical systems, apparently independent scaling exponents turn out to be related by exact identities. A striking recent example comes from the theory of jamming: collections of particles, grains, or spheres become rigid at a critical packing density, and the resulting marginal state exhibits power-law behavior in quantities such as gaps, forces, soft modes, and avalanches.

In the full replica symmetry breaking (fullRSB) description of jamming, one obtains an exact relation of the schematic form

$$𝑎+𝑏=1,$$

where $𝑎$ and $𝑏$ are critical exponents associated with different asymptotic sectors of the theory. The important point is that this identity is not merely a numerical coincidence. It arises from the combination of:

1. a hierarchical organization of metastable states;
2. a marginal stability condition;
3. scaling ansätze for the relevant fullRSB functions.

This suggests a broader question:

$$\boxed{{\displaystyle \text{Can marginality in a hierarchical system force exact relations among scaling exponents?}}}$$

If the answer is yes, then fullRSB jamming may be one realization of a more general mechanism: ultrametric marginality.

The purpose of this article is to describe a controlled mathematical laboratory for exploring that possibility: a Fano-restricted scalar field theory over the 7-adic numbers ${\mathbb{𝑄}}_7$, together with its interpretation on the Bruhat–Tits tree ${𝑇}_7$.

We do not claim that this construction is a literal model of jamming. Rather, we regard it as a toy framework in which hierarchy, ultrametricity, spectral marginality, and exact combinatorics coexist in a solvable setting.

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2. Why p-adic geometry?

The fullRSB formalism is intrinsically hierarchical. States are organized into clusters of clusters, and the resulting structure is ultrametric. An ultrametric distance satisfies

$$𝑑(𝑥,𝑧)\le \max \{𝑑(𝑥,𝑦),𝑑(𝑦,𝑧)\},$$

which is stronger than the ordinary triangle inequality. Geometrically, this means that balls are either disjoint or nested.

The field of $𝑝$-adic numbers ${\mathbb{𝑄}}_{𝑝}$ provides one of the cleanest mathematical realizations of ultrametric geometry. The $𝑝$-adic norm

$$\mathrm{\mid }𝑥{\mathrm{\mid }}_{𝑝}={𝑝}^{-{𝑣}_{𝑝}(𝑥)}$$

defines an ultrametric distance, and the resulting space can be represented as the boundary of a tree.

For $𝑝=7$, the relevant tree is the Bruhat–Tits tree ${𝑇}_7$. Schematically,

$$\mathrm{\partial }{𝑇}_7\simeq {\mathbb{𝑃}}^1({\mathbb{𝑄}}_7)={\mathbb{𝑄}}_7\cup \{\mathrm{\infty }\}\mathrm{.}$$

The radial direction of the tree corresponds to scale, while the boundary points correspond to microscopic or asymptotic degrees of freedom.

This gives a natural dictionary:

$$\text{hierarchical scale}⟷\text{depth in }{𝑇}_7,$$

$$\text{boundary observables}⟷\text{fields or operators on }{\mathbb{𝑄}}_7\mathrm{.}$$

The attraction of the $𝑝$-adic setting is therefore not merely aesthetic. It provides an explicit geometric arena in which hierarchical renormalization, ultrametric correlations, and marginal operators can be studied with unusual mathematical control.

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3. A solvable toy model: Fano-restricted cubic theory on ${\mathbb{𝑄}}_7$

The starting point is a cubic scalar theory over ${\mathbb{𝑄}}_7$ with seven scalar fields

$${\mathrm{\Phi }}_{𝑎}(𝑥),𝑎=1,\dots ,7,$$

whose interaction is restricted by the incidence structure of the Fano plane $\mathrm{P}\mathrm{G}(2,2)$.

The Fano plane is the smallest projective plane: it has 7 points and 7 lines, each line contains 3 points, and each point lies on 3 lines. Its automorphism group is

$$\mathrm{Aut}(\mathrm{P}\mathrm{G}(2,2))\simeq \mathrm{P}\mathrm{S}\mathrm{L}(2,7),$$

of order 168.

We define a totally symmetric incidence tensor

$${𝑊}_{𝑎𝑏𝑐}=\{\begin{array}{ll}1,& \{𝑎,𝑏,𝑐\}\text{ is a Fano line},\\ 0,& \text{otherwise}\mathrm{.}\end{array}$$

The interaction is

$${𝑆}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{{𝜆}_3}{3!}{\int }_{{\mathbb{𝑄}}_7}\sum _{𝑎,𝑏,𝑐=1}^7{𝑊}_{𝑎𝑏𝑐}{\mathrm{\Phi }}_{𝑎}(𝑥){\mathrm{\Phi }}_{𝑏}(𝑥){\mathrm{\Phi }}_{𝑐}(𝑥){𝑑}^7𝑥\mathrm{.}$$

The free kinetic term is governed by the Vladimirov fractional derivative ${𝐷}^{𝛼}$, whose momentum-space symbol is

$$\widehat{{𝐷}^{𝛼}\mathrm{\Phi }}(𝑘)=\mathrm{\mid }𝑘{\mathrm{\mid }}_7^{𝛼}\widehat{\mathrm{\Phi }}(𝑘)\mathrm{.}$$

The free propagator in momentum space is therefore

$${\widetilde{𝐺}}_0(𝑘)=\mathrm{\mid }𝑘{\mathrm{\mid }}_7^{-𝛼}\mathrm{.}$$

The parameter $𝛼\in (0,1)$ controls the kinetic scaling and will later be promoted to a deformation parameter.

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4. Exact one-loop structure

One of the main features of this toy model is that its one-loop combinatorics can be computed exactly and independently verified by brute-force enumeration.

4.1 Vertex correction

The one-loop triangle correction to the cubic vertex receives three independent contributions:

1. a Fano incidence factor,
2. a Wick symmetry factor,
3. a $𝑝$-adic loop integral residue.

The Fano incidence factor is

$$𝑇=5.$$

The Wick symmetry factor for the triangle topology is

$${𝜅}_{𝑊}=1.$$

The relevant triangle integral is

$$𝐶(𝛼)={\int }_{{\mathbb{𝑄}}_7}\mathrm{\mid }𝑥{\mathrm{\mid }}_7^{-𝛼}\mathrm{\mid }𝑥-1{\mathrm{\mid }}_7^{-𝛼}\mathrm{\mid }𝑥-2{\mathrm{\mid }}_7^{-𝛼}{𝑑}^7𝑥\mathrm{.}$$

This integral has a simple ultraviolet pole at

$$𝛼=\frac{1}{3},$$

with residue

$${\mathrm{Res}}_{𝛼=1\mathrm{/}3}𝐶(𝛼)=\frac{2}{7\ln 7}\mathrm{.}$$

Therefore the exact vertex coefficient is

$${𝑐}_3=𝑇{𝜅}_{𝑊}{\mathrm{Res}}_{𝛼=1\mathrm{/}3}𝐶(𝛼)=5\cdot 1\cdot \frac{2}{7\ln 7}=\frac{10}{7\ln 7}\mathrm{.}$$

This value is exact within the model.

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4.2 Self-energy correction

The one-loop bubble correction to the propagator is governed by a different combinatorial factor and a different marginal value of $𝛼$.

The generic Wick symmetry factor for the bubble diagram is $1\mathrm{/}2$. The Fano flavor factor for fixed external flavor is 6. Hence the combined bubble factor is

$${𝐹}_{\mathrm{b}\mathrm{u}\mathrm{b}\mathrm{b}\mathrm{l}\mathrm{e}}=\frac{1}{2}\cdot 6=3.$$

The bubble integral is

$${𝐼}_{\mathrm{b}\mathrm{u}\mathrm{b}}(𝛼)={\int }_{{\mathbb{𝑄}}_7}\mathrm{\mid }𝑘{\mathrm{\mid }}_7^{-𝛼}\mathrm{\mid }𝑘-1{\mathrm{\mid }}_7^{-𝛼}{𝑑}^7𝑘\mathrm{.}$$

It has a simple pole at

$$𝛼=\frac{1}{2},$$

with residue

$${\mathrm{Res}}_{𝛼=1\mathrm{/}2}{𝐼}_{\mathrm{b}\mathrm{u}\mathrm{b}}(𝛼)=\frac{3}{7\ln 7}\mathrm{.}$$

Thus the total self-energy coefficient is

$${𝐹}_{\mathrm{b}\mathrm{u}\mathrm{b}\mathrm{b}\mathrm{l}\mathrm{e}}{\mathrm{Res}}_{𝛼=1\mathrm{/}2}{𝐼}_{\mathrm{b}\mathrm{u}\mathrm{b}}=3\cdot \frac{3}{7\ln 7}=\frac{9}{7\ln 7}\mathrm{.}$$

The appearance of two distinct marginal values,

$${𝛼}_1=\frac{1}{3},{𝛼}_2=\frac{1}{2},$$

is not an inconsistency. The two diagrams have different loop topologies: the vertex correction involves three propagators, while the self-energy involves two. Different topologies naturally select different scaling thresholds.

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5. Beta function and complex fixed point

The classical scaling dimension of the cubic coupling is

$$[{𝜆}_3]=\frac{3𝛼-1}{2}\mathrm{.}$$

Therefore the coupling is classically marginal at

$$𝛼=\frac{1}{3}\mathrm{.}$$

Promoting $𝛼$ to a continuous deformation parameter, the one-parameter beta function in the simplest one-loop truncation is

$$𝛽({𝜆}_3;𝛼)={𝑎}_1(𝛼){𝜆}_3-{𝑐}_3{𝜆}_3^3,$$

with

$${𝑎}_1(𝛼)=-\frac{3𝛼-1}{2}\mathrm{.}$$

The nontrivial fixed points satisfy

$$({𝜆}_3^{\ast }{)}^2=\frac{{𝑎}_1(𝛼)}{{𝑐}_3}\mathrm{.}$$

Since

$${𝑐}_3=\frac{10}{7\ln 7}>0,$$

the nature of the fixed point is controlled by the sign of ${𝑎}_1(𝛼)$.

For

$$𝛼\in (\frac{1}{3},\frac{1}{2}],$$

we have

$${𝑎}_1(𝛼)<0.$$

Therefore the nontrivial fixed point is purely imaginary:

$${𝜆}_3^{\ast }=\pm 𝑖\sqrt{\frac{\mathrm{\mid }{𝑎}_1(𝛼)\mathrm{\mid }}{{𝑐}_3}}\mathrm{.}$$

Within this truncation, the theory does not possess a real interacting fixed point in the interval between the two marginal points. Instead, the complex fixed point organizes a slow, walking-type regime in the complexified coupling plane.

This behavior is conceptually related to the walking phenomena studied in complex CFTs and Yang–Lee-type systems, although the precise physical interpretation in the present $𝑝$-adic toy model remains an open question.

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6. External sources and negotiated minima

A natural extension is to couple the isolated system to an external environment. In field-theoretic language, one introduces a source $𝐽$ and studies the effective action

$$\mathrm{\Gamma }[{\mathrm{\Phi }}_{\mathrm{c}\mathrm{l}},𝐽]\mathrm{.}$$

The conceptual point is important:

$$\boxed{{\displaystyle \text{For }𝐽\mathrm{\ne }0,\text{ the energy minimum is not intrinsic to the isolated system.}}}$$

Instead, the physical state is co-determined by the internal dynamics and the external source.

In the present program, one possible implementation uses the unique unramified quadratic extension

$${\mathbb{𝑄}}_{49}\mathrm{/}{\mathbb{𝑄}}_7\mathrm{.}$$

The trace map

$${\mathrm{Tr}}_{{\mathbb{𝑄}}_{49}\mathrm{/}{\mathbb{𝑄}}_7}(𝑎+𝑏𝜔)=2𝑎$$

leads to directional asymmetries between extension and restriction kernels in Fourier space. These asymmetries are supported on what may loosely be called the “imaginary” shells of ${\mathbb{𝑄}}_{49}$.

A candidate external-source self-energy has been proposed in this framework. It predicts that the source acts primarily as an infrared regulator, shifting the linear coefficient of the beta function while preserving the complex nature of the fixed point. In that scenario, the external source does not stabilize the system into a conventional real fixed point; it accelerates the walking regime.

However, this part of the program is still being refined. In particular, the precise shell-volume normalization in the source integral must be treated carefully, because the standard $𝑝$-adic shell volume satisfies

$$\mathrm{vol}\{𝑥\in {\mathbb{𝑄}}_7:\mathrm{\mid }𝑥{\mathrm{\mid }}_7=7^{𝑛}\}=\frac{6}{7}{7}^{𝑛}\mathrm{.}$$

Any external-source calculation involving large shells must be checked against this measure convention. The conceptual idea—that environmental coupling can shift marginality without removing it—remains one of the interesting directions of the program.

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7. Non-local interactions and the holographic turn

In addition to the Local construction with seven flavor fields, one can define a Non-Local construction in which a single scalar field interacts with itself through Fano-compatible residue classes:

$${𝑆}_{\mathrm{N}\mathrm{L}}=\frac{{𝜆}_3^{\mathrm{N}\mathrm{L}}}{7}\int \mathrm{\Phi }(𝑥)\mathrm{\Phi }(𝑦)\mathrm{\Phi }(𝑧){𝛿}_{\mathrm{F}\mathrm{a}\mathrm{n}\mathrm{o}}(𝑥7,𝑦7,𝑧7){𝑑}^7𝑥{𝑑}^7𝑦{𝑑}^7𝑧\mathrm{.}$$

Dimensional analysis gives

$$[{𝜆}_3^{\mathrm{N}\mathrm{L}}]=\frac{3(1+𝛼)}{2}\mathrm{.}$$

For $𝛼\in (0,1)$, this is strictly positive:

$$[{𝜆}_3^{\mathrm{N}\mathrm{L}}]>0.$$

Therefore the Non-Local coupling is always relevant and has no natural marginal point without additional fine-tuning.

This negative result is actually conceptually useful. It suggests that the Non-Local interaction should not be interpreted purely as a boundary perturbation. Instead, non-locality on the boundary may be interpreted as the image of propagation through the bulk of the Bruhat–Tits tree.

In other words,

$$\boxed{{\displaystyle \text{boundary non-locality}⟷\text{bulk geometry}\mathrm{.}}}$$

This is the point at which the construction begins to resemble a toy version of holography.

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8. The Bruhat–Tits tree as a geometric laboratory

The Bruhat–Tits tree ${𝑇}_7$ is a $(7+1)$-regular infinite tree associated with $\mathrm{P}\mathrm{G}\mathrm{L}(2,{\mathbb{𝑄}}_7)$. Its boundary is

$$\mathrm{\partial }{𝑇}_7\simeq {\mathbb{𝑃}}^1({\mathbb{𝑄}}_7)\mathrm{.}$$

In this setting, one can define a bulk-to-boundary propagator. For a boundary operator of scaling dimension $\mathrm{\Delta }$, the two-point coefficient takes the form

$${𝐶}_{\mathrm{\Delta }}=\frac{1-7^{-(2\mathrm{\Delta }+1)}}{1-7^{-2\mathrm{\Delta }}}\mathrm{.}$$

At the special value

$$\mathrm{\Delta }=\frac{1}{2},$$

one obtains the exact rational value

$${𝐶}_{1\mathrm{/}2}=\frac{1-7^{-2}}{1-7^{-1}}=\frac{48\mathrm{/}49}{6\mathrm{/}7}=\frac{8}{7}\mathrm{.}$$

This is striking because

$$\frac{8}{7}=\frac{7+1}{7},$$

which is directly related to the valency of the tree ${𝑇}_7$.

A discrete analogue of the Ryu–Takayanagi relation can also be proposed:

$$𝑆(𝐴,𝐵)=\frac{𝑐}{6}{𝑛}_{𝐴𝐵},$$

where ${𝑛}_{𝐴𝐵}$ is a discrete measure of hierarchical separation between boundary regions $𝐴$ and $𝐵$, and $𝑐$ is an effective central charge.

At present, this holographic layer is exploratory. It is best regarded as a structural framework rather than an established duality. Nevertheless, it provides a natural language for thinking about how hierarchical bulk geometry can encode boundary correlations.

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9. From structure to properties

Initially, one might try to identify the Bruhat–Tits tree directly with the fullRSB state space. That is probably too strong.

A more promising interpretation is this:

$$\boxed{{\displaystyle \text{The tree does not model the microscopic contact network.}}}$$

Instead,

$$\boxed{{\displaystyle \text{The tree models the hierarchical organization of observable properties.}}}$$

In jamming, the relevant observables include forces, gaps, soft modes, susceptibilities, and response functions. These are not necessarily organized by the literal geometry of the particles. They may instead be organized by an effective hierarchy of scales.

In this view, the Bruhat–Tits tree is not a literal physical network. It is a spectral and geometric device for encoding how properties correlate across scales.

For example, one could imagine two property-like observables:

$$𝐹\sim \text{force-like channel},𝐻\sim \text{gap-like channel}\mathrm{.}$$

Their correlations might scale as

$${𝐶}_{𝐹}(𝑛)\sim 7^{-𝑛{\mathrm{\Delta }}_{𝐹}},{𝐶}_{𝐻}(𝑛)\sim 7^{-𝑛{\mathrm{\Delta }}_{𝐻}},$$

where $𝑛$ is depth in the tree.

If marginal stability imposes a condition such as

$${𝐶}_{𝐹}(𝑛){𝐶}_{𝐻}(𝑛)\sim 7^{-𝑛},$$

then one obtains

$${\mathrm{\Delta }}_{𝐹}+{\mathrm{\Delta }}_{𝐻}=1.$$

This is only a schematic example, but it illustrates the kind of mechanism we are looking for:

$$\text{hierarchical correlations}+\text{marginal spectral condition}\Rightarrow \text{exponent identity}\mathrm{.}$$

This shift from structure to properties is crucial. It avoids the need to claim that jamming is literally $𝑝$-adic. Instead, the claim becomes more modest and more testable:

$$\boxed{{\displaystyle 𝑝\text{-adic trees may provide effective geometries for marginal property correlations.}}}$$

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10. Weighted trees, channels, and emergent ultrametricity

A further generalization is to regard the tree as an emergent object generated by weighted interaction channels.

One may imagine a hierarchical system with internal and external weights:

$${𝑊}_{\mathrm{i}\mathrm{n}\mathrm{t}},{𝑊}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{.}$$

A schematic recursion for two coupled channels could be

$$\left(\begin{array}{c}{𝐹}_{𝑛+1}\\ {𝐻}_{𝑛+1}\end{array}\right)=𝑀\left(\begin{array}{c}{𝐹}_{𝑛}\\ {𝐻}_{𝑛}\end{array}\right),$$

with

$$𝑀=\left(\begin{array}{cc}{𝑤}_{𝐹}& {𝑢}_{𝐹}\\ {𝑢}_{𝐻}& {𝑤}_{𝐻}\end{array}\right)\mathrm{.}$$

Here ${𝑤}_{𝐹},{𝑤}_{𝐻}$ are internal weights, while ${𝑢}_{𝐹},{𝑢}_{𝐻}$ are cross-couplings. External sources can shift the entries of $𝑀$.

Marginality could then correspond to a spectral condition such as

$${𝜆}_{\max }(𝑀)=1.$$

If the eigenvalues of $𝑀$ control the scaling dimensions of the channels, then a marginal eigenvalue may impose a relation among the corresponding exponents.

This type of construction is intentionally simple. It resembles a hierarchical neural network, a tensor-network renormalization scheme, or a multichannel RG flow. The point is not to model a neuron or a granular packing literally. The point is to isolate the minimal mechanism by which hierarchical weights generate marginal scaling laws.

In this interpretation, the $𝑝$-adic tree is not fundamental. It is an emergent coordinate system for scale:

$$\boxed{{\displaystyle 𝑝\text{-adic discreteness as an effective description of hierarchical resolution.}}}$$

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11. What would count as real evidence?

For this program to move beyond analogy, several conditions must be met.

11.1. A derived exponent identity

One must derive an identity such as

$${\mathrm{\Delta }}_{𝐹}+{\mathrm{\Delta }}_{𝐻}=1$$

from explicit equations, not from numerical coincidence or post hoc reinterpretation.

11.2. A marginal operator

There should be a clearly defined spectral operator whose marginal mode imposes the identity. In fullRSB language, the natural candidate is related to the replicon. In the $𝑝$-adic toy model, one needs an analogue of that operator.

11.3. Universality tests

The construction should be varied:

$$𝑝=7\to 𝑝=5,11,13,\dots$$

and

$$\mathrm{P}\mathrm{G}(2,2)\to \mathrm{P}\mathrm{G}(2,𝑞)$$

or other incidence structures.

If the exponent identity survives these changes, it may reflect a universal mechanism. If it disappears, then the identity was specific to the chosen toy model.

11.4. A clean distinction between established and conjectural results

The current status is roughly the following:

Status	Content
Established	Exact Fano combinatorics, one-loop residues, complex fixed point in the one-parameter beta function
Established but limited	The Local toy model with $𝑝=7$ and Fano incidence
Exploratory	External-source acceleration, detailed walking interpretation
Exploratory	Holographic interpretation via ${𝑇}_7$
Conjectural	Discrete Ryu–Takayanagi relation, central charge formula, direct connection to jamming exponents

Maintaining this distinction is essential.

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12. Relation to fullRSB jamming

The original motivation comes from fullRSB jamming, where marginality produces exact exponent relations. However, the present construction should not be oversold.

At present, the $𝑝$-adic/Fano model does not contain:

• a replica limit;
• an overlap matrix ${𝑄}_{𝑎𝑏}$;
• a fully defined replicon operator;
• a derivation of the Parisi equation;
• a direct mapping between jamming exponents and $𝑝$-adic scaling dimensions.

Therefore, the connection to jamming remains hypothetical.

The more defensible claim is:

$$\boxed{{\displaystyle \text{The }𝑝\text{-adic construction is a laboratory for hierarchical marginality,}}}$$

not

$$\boxed{{\displaystyle \text{The }𝑝\text{-adic construction is a model of jamming.}}}$$

This distinction is not a weakness. It is what keeps the program scientifically honest.

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13. The central hypothesis

The research program can be summarized by a single working hypothesis:

$$\boxed{{\displaystyle \text{Ultrametric hierarchy}+\text{marginal spectral mode}\Rightarrow \text{universal scaling constraint}\mathrm{.}}}$$

In the jamming/fullRSB realization, the constraint is schematically

$$𝑎+𝑏=1.$$

In the $𝑝$-adic/Fano laboratory, the goal is to determine whether an analogous constraint can be derived from explicit tree-based spectral equations.

If such a derivation can be found, then identities like $𝑎+𝑏=1$ may be understood not as isolated facts about particle packings, but as manifestations of a broader principle governing hierarchical marginal systems.

If no such derivation can be found, the construction remains valuable as an exactly solvable toy model of non-Archimedean field theory, but the bridge to jamming should be regarded as unsuccessful.

Either outcome would be informative.

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14. Conclusion

The Fano-restricted $𝑝$-adic construction provides a rare combination:

• exact combinatorics;
• explicit $𝑝$-adic loop integrals;
• controllable RG deformation;
• complex fixed points and walking behavior;
• a natural tree geometry;
• a possible holographic interpretation.

These ingredients make it a useful laboratory for exploring a deeper question:

$$\boxed{{\displaystyle \text{Is marginality intrinsically hierarchical?}}}$$

The most promising direction is not to force the toy model to be a literal theory of jamming. Instead, the model should be used to study how hierarchical weights, spectral marginality, and ultrametric correlations can generate exponent identities.

The next step is therefore not to add more decoration, but to reduce the mechanism to its simplest form:

1. define a weighted hierarchical system;
2. identify two scaling observables;
3. impose a marginal spectral condition;
4. derive an exponent relation;
5. test whether the relation survives changes of geometry and prime.

If this minimal program succeeds, then the present construction will have done more than produce elegant mathematics. It will have identified a general mechanism by which marginal hierarchical systems constrain their own scaling behavior.