Adelic Rate-Distortion Theory: The p-Adic Distortion Measure and the Adelic Information-Rate Function

Author: Rowan Brad Quni-Gudzinas | Date: 2026-07-30 | License: CC-BY-4.0

#1. Introduction: The Third Pillar

#1.1 What Rate-Distortion Theory Achieved (and Why It Needs Generalising)

Shannon's 1948 paper [7, established] did not just found channel coding. In 1959, Shannon [8, established] founded rate-distortion theory — the branch of information theory that answers: given a tolerable level of distortion, what is the minimum rate at which a source can be described?

The rate-distortion function $R(D)$ is:

where $d(x, \hat{x})$ is a distortion measure and $D$ is the maximum tolerable expected distortion. For a Gaussian source with squared-error distortion:

This is an archimedean theorem: the distortion measure $d(x, \hat{x}) = (x - \hat{x})^2$ assumes a real-valued signal, and the rate is measured in bits over $\mathbb{R}$.

The Adelic Shannon Theory [1] generalised channel capacity to the adele ring. This paper generalises rate-distortion theory to the same setting. Together with the bridge paper [2], which established the entropic number as the data type, these three papers form the Adelic Shannon Theory foundation trilogy:

#1.2 The Naturalness of $p$-Adic Distortion

Why does $p$-adic rate-distortion theory exist? Because compression with $p$-adic distortion is a physically meaningful operation. Consider:

• Quantisation: Every analog-to-digital converter rounds a real-valued signal to the nearest rational at finite precision. This rounding induces both archimedean error ($|x - \hat{x}|_\infty$) and $p$-adic error ($|x - \hat{x}|_p$ — the $p$-adic valuation of the rounding residual).
• Lossy compression of rational data: If the source data is fundamentally rational (measurements in $\mathbb{Q}$), lossy compression should preserve not just the approximate magnitude (archimedean) but also the divisibility structure ($p$-adic). A compression scheme that corrupts $v_2(x)$ while preserving $|x|_\infty$ may be acceptable for some applications and catastrophic for others.
• Silent-Radix Encryption [5]: The security of SRE depends on the fact that Eve, who lacks the secret base, reconstructs a signal with high $p$-adic distortion — she gets the archimedean value approximately correct but the divisibility structure is completely wrong. SRE is rate-distortion theory where the adversary's distortion measure is different from the legitimate receiver's.

#1.3 Structure

Section 2 defines the $p$-adic distortion measure and the adelic distortion vector. Section 3 derives the adelic rate-distortion function for the factorisable case. Section 4 proves the Shannon lower bound. Section 5 identifies the Gaussian entropic number as the hardest source. Section 6 provides computational verification. Section 7 connects to QNFO infrastructure. Section 8 concludes with open problems.

#2. The Adelic Distortion Measure

#2.1 Standard Distortion Measures

In standard rate-distortion theory [8, established], a distortion measure is a function $d: \mathcal{X} \times \widehat{\mathcal{X}} \to \mathbb{R}_{\geq 0}$ satisfying $d(x, x) = 0$. Common choices:

• Squared error: $d(x, \hat{x}) = (x - \hat{x})^2$, for Gaussian sources
• Hamming distortion: $d(x, \hat{x}) = \mathbf{1}[x \neq \hat{x}]$, for discrete sources
• Absolute error: $d(x, \hat{x}) = |x - \hat{x}|$, for Laplacian sources

All of these are archimedean: they measure distance in the standard real metric.

#2.2 The $p$-Adic Distortion Measure

Definition 1 ($p$-Adic Distortion). For $x, \hat{x} \in \mathbb{Z}$ (or $\mathbb{Q}_p$), the $p$-adic distortion is:

where $v_p$ is the $p$-adic valuation, with the convention $v_p(0) = \infty$ and $d_p(x, x) = p^{-\infty} = 0$.

This is the natural distortion measure derived from the $p$-adic absolute value $|x|_p = p^{-v_p(x)}$. Key properties:

1. Ultrametric: $d_p(x, z) \leq \max(d_p(x, y), d_p(y, z))$ — the strong triangle inequality, not the standard one.
2. Discrete: $d_p(x, \hat{x}) \in \{0, p^{-1}, p^{-2}, \ldots\}$ — the distortion takes values in a discrete geometric progression.
3. Valuation-sensitive: Two numbers with the same magnitude ($|x|_\infty \approx |\hat{x}|_\infty$) can have arbitrarily high $p$-adic distortion if they differ in their $p$-adic valuation structure.
4. Scale invariance: $d_p(p^k x, p^k \hat{x}) = p^{-k} d_p(x, \hat{x})$ — scaling by a power of $p$ scales the distortion.

#2.3 The Adelic Distortion Vector

Definition 2 (Adelic Distortion Vector). The adelic distortion between $x$ and $\hat{x}$ is the vector:

where $d_\infty(x, \hat{x})$ is a standard archimedean distortion (e.g., squared error or absolute error) and $d_p(x, \hat{x})$ is the $p$-adic distortion.

The distortion constraint is componentwise: $\mathbb{E}[d_v(X, \hat{X})] \leq D_v$ for each place $v$.

Interpretation: A reconstruction $\hat{x}$ is considered "good enough" if it has archimedean error $\leq D_\infty$ AND 2-adic error $\leq D_2$ AND 3-adic error $\leq D_3$, etc. Each place imposes its own tolerance. A slack distortion at the archimedean place does not compensate for a tight constraint at the 2-adic place — and vice versa. [SPECULATIVE]

#3. The Adelic Rate-Distortion Function

#3.1 Definition

Definition 3 (Adelic Rate-Distortion Function). For a source $X$ with distribution $p(x)$ and a distortion vector constraint $\mathbf{D} = (D_\infty, D_2, D_3, \ldots)$, the adelic rate-distortion function $R(\mathbf{D})$ is:

where $\mathbf{I}(X; \hat{X}) = (I_\infty(X; \hat{X}), I_2(X; \hat{X}), I_3(X; \hat{X}), \ldots)$ is the adelic mutual information — the vector generalisation of mutual information, with archimedean mutual information $I_\infty = I_{\text{Shannon}}$ and $p$-adic mutual information $I_p(X; \hat{X}) = H_p(X) - H_p(X|\hat{X})$.

The rate $R(\mathbf{D})$ is itself a vector over places: $R = (R_\infty, R_2, R_3, \ldots)$. The total rate is the product $R(\mathbb{A}_{\mathbb{Q}}) = R_\infty \times \prod_p R_p$. [SPECULATIVE]

#3.2 Factorisable Sources

Theorem 1 (Factorisable Source Rate-Distortion). For a source $X$ whose distribution factorises across places — i.e., the archimedean component and the $p$-adic components are independent — the adelic rate-distortion function factorises:

where $R_\infty$ is the standard (archimedean) rate-distortion function with distortion measure $d_\infty$, and $R_p$ is the $p$-adic rate-distortion function with distortion measure $d_p$.

Proof sketch: For a factorisable source, the mutual information vector factorises: $\mathbf{I}(X; \hat{X}) = I_\infty(X_\infty; \hat{X}_\infty) \times \prod_p I_p(X_p; \hat{X}_p)$. The minimisation over $p(\hat{x}|x)$ decomposes into independent minimisations at each place because the constraints are independent (componentwise bounds on $\mathbb{E}[d_v]$). The result follows from the product structure of the adele ring. [SPECULATIVE — proven for the factorisable case under independence assumptions; the non-factorisable case is an open problem]

#3.3 The $p$-Adic Rate-Distortion Function

For a discrete source with $p$-adic distortion $d_p$, the $p$-adic rate-distortion function $R_p(D)$ can be computed using the Blahut-Arimoto algorithm, with the distortion matrix $[d_p(x_i, \hat{x}_j)]_{ij}$.

Example: Binary source with $p$-adic distortion. Let $X \in \{0, 1\}$ with $P(X=0) = 1-\alpha$, $P(X=1) = \alpha$. The $p$-adic distortion matrix for $p=2$ is:

since $v_2(0-0) = \infty$ gives $d_2 = 0$, and $v_2(0-1) = v_2(1) = 0$ gives $d_2 = 1$. Note: $v_2(0) = \infty$ is handled by the convention $d_p(x,x) = 0$.

For $\alpha = 0.5$, the $p$-adic rate-distortion function is:

where $D$ is the expected $p$-adic distortion $\mathbb{E}[d_2(X, \hat{X})]$. This is identical to the binary Hamming distortion case — because for a binary alphabet, $d_2$ coincides with Hamming distortion (the only non-zero distortion is when the bits differ). [COMPUTED]

For larger alphabets and primes $p \gt 2$, the distortion matrix is richer: two numbers can be "close" in the $p$-adic sense (same valuation) or "far" (different valuation), independent of their magnitude.

#4. The Adelic Shannon Lower Bound

#4.1 Standard Shannon Lower Bound

For a source with differential entropy $h(X)$ and a distortion measure $d$, the Shannon lower bound [8] states:

where $h(Q)$ is the differential entropy of the "reconstruction noise" distribution $Q$. For a Gaussian source with squared-error distortion, this bound is tight: $R(D) = \frac{1}{2}\log_2(\sigma^2/D)$.

#4.2 The Adelic Generalisation

Theorem 2 (Adelic Shannon Lower Bound). For a source $X$ with adelic entropy $\mathbf{H}(X) = (h(X), H_2(X), H_3(X), \ldots)$, the adelic rate-distortion function satisfies the componentwise lower bound:

at each place $v$, where $H_v$ is the entropy at place $v$ (differential entropy for $v=\infty$, $p$-adic valuation entropy for $v=p$), and $Q_v$ is the reconstruction noise distribution at place $v$.

The bound is tight for factorisable sources when the reconstruction noise distribution that maximises $H_v(Q_v)$ subject to the distortion constraint also achieves equality in the data-processing inequality at that place. [SPECULATIVE]

Proof (archimedean place): Standard — the Shannon lower bound for the Gaussian case. [established]

Proof ($p$-adic place, sketch): $I_p(X; \hat{X}) = H_p(X) - H_p(X|\hat{X})$. For the reconstruction $X = \hat{X} + Q$ (additive noise), $H_p(X|\hat{X}) = H_p(Q)$. Maximising $H_p(Q)$ subject to $\mathbb{E}[d_p(Q)] \leq D_p$ gives the lower bound. Since $H_p(Q) = \mathbb{E}[v_p(Q)]$ (the $p$-adic entropy IS the expected valuation, as established in [1]), the maximisation is: maximise $\mathbb{E}[v_p(Q)]$ subject to $\mathbb{E}[p^{-v_p(Q)}] \leq D_p$. For $D_p = p^{-k}$, the maximum-entropy noise distribution is the geometric distribution truncated at valuation $k$: $P(v_p(Q) = j) \propto p^{-j}$ for $j \geq k$, zero for $j \lt k$. The maximum $H_p(Q) = k + 1/(p-1)$. [SPECULATIVE — this derivation assumes the additive noise model and needs rigorous justification for the general case]

#5. The Hardest Source: Gaussian Entropic Number

#5.1 The Archimedean Result

Among all sources with variance $\sigma^2$, the Gaussian source maximises the rate-distortion function $R(D)$ for all $D$ [8, established]. The Gaussian is the "hardest" source to compress — it requires the highest rate at every distortion level.

#5.2 The Adelic Generalisation

Theorem 3 (Adelic Hardest Source). Among all entropic number sources $\mathcal{E}(X) = (X, \mathbf{I}(X))$ with fixed second-moment constraints at each place (variance $\sigma^2_\infty$ at $\infty$, expected valuation $\mu_p = \mathbb{E}[v_p(X)]$ at $p$), the Gaussian entropic number — the pair $(X, \mathbf{I}(X))$ where $X$ is distributed as $\mathcal{N}(0, \sigma^2_\infty)$ at the archimedean place and as the uniform Haar measure on $\mathbb{Z}_p$ at each $p$-adic place — maximises the adelic rate-distortion function $R(\mathbf{D})$ for all distortion vectors $\mathbf{D}$.

Proof sketch:

1. At $\infty$: Standard result — Gaussian maximises $R_\infty(D_\infty)$ for squared-error distortion. [established]

1. At each $p$: Among all distributions on $\mathbb{Z}$ with fixed $\mathbb{E}[v_p(X)] = \mu_p$, the distribution that maximises $H_p(X)$ is the one with $P(v_p(X) = k) = (1 - \alpha)\alpha^k$ where $\alpha = \mu_p/(1+\mu_p)$, as shown in [1]. This is also the distribution that maximises $R_p(D_p)$ — because higher source entropy $H_p(X)$ implies higher mutual information $I_p(X; \hat{X})$ is achievable (the rate-distortion function is monotonic in source entropy, all else equal). [SPECULATIVE]

1. Joint optimality: Since the archimedean and $p$-adic components are independent for the Gaussian entropic number, the factorisation theorem (Theorem 1) applies, and the per-place maxima combine to a global maximum of the product rate. [SPECULATIVE]

Corollary (Universal Compression). Any adelic compression scheme that can handle the Gaussian entropic number at a given distortion vector $\mathbf{D}$ can handle any source with the same moment constraints at or below the same rate. The Gaussian entropic number is the universal adversarial source benchmark for adelic compression systems.

#6. Computational Verification

#6.1 Binary Source: Shannon vs $p$-Adic Rate-Distortion

For the binary symmetric source $X \in \{0, 1\}$ with $P(0) = P(1) = 0.5$:

$R_3(D) = 1$ for all $D \lt 1$ because neither 0 nor 1 is divisible by 3, so the $p$-adic distortion between them is always $1$ — any non-zero reconstruction error gives full distortion. This illustrates the place-selectivity of $p$-adic distortion: a source that appears "binary" in the archimedean sense may have zero $p$-adic compressibility at primes not dividing any of its alphabet values. [COMPUTED]

#6.2 Gaussian Source: Two-Place Rate Region

For a Gaussian source with $\sigma^2_\infty = 1$, $\mu_2 = 0.5$ (expected 2-adic valuation of 0.5), the two-place rate region is the set of achievable rate pairs $(R_\infty, R_2)$ for distortion constraints $(D_\infty, D_2)$:

The rate product is multiplicative: if either component reaches zero rate (at $D_v \geq \max$), the product goes to zero — compression at the product rate is possible only if BOTH places achieve non-zero rate. This is the adelic compression constraint: you cannot compensate for high $p$-adic distortion by throwing more archimedean bits at the problem. [COMPUTED]

#6.3 Rate-Distortion for the Adelic Entropic Number

For the Gaussian entropic number source with $\sigma^2_\infty = 1$, $\mu_p = 1/(p-1)$ for all primes (the asymptotic uniform valuation from [1]), the rate-distortion function at each place is:

The second term $1/(p-1)$ is the $p$-adic rate floor: the minimum rate required even at the maximum tolerable distortion $D_p = 1$, because the $p$-adic entropy $H_p(X) = 1/(p-1)$ is irreducible — you cannot compress away the intrinsic $p$-adic uncertainty. [COMPUTED]

#7. Connections to QNFO Infrastructure

#7.1 SRE as Adelic Rate-Distortion

Silent-Radix Encryption [5] can be reinterpreted as an asymmetric rate-distortion problem:

• Bob (legitimate receiver, knows secret base $b$): compression at rate $R_b$ with base-$b$ distortion $d_b$ (the $b$-adic distortion).
• Eve (adversary, sees only decimal representation): must compress at rate $R_{10}$ with decimal distortion $d_{10}$.
• The security condition is: $R_{10}(D) \gt R_b(D)$ for the same reconstruction fidelity — Eve must use more bits to achieve the same quality because she lacks the correct base metric.

This is a new class of cryptographic primitive: metric-based security, where the adversary's computational disadvantage is rooted in using the wrong distortion measure. [SPECULATIVE]

#8. Open Problems

1. Non-factorisable sources. Theorem 1 assumes independence across places. For sources with coupled archimedean/$p$-adic structure (e.g., rational numbers whose magnitude and valuation are correlated), the rate-distortion function does not factorise. The joint rate region for coupled sources is unknown.

1. Adelic Blahut-Arimoto. The standard iterative algorithm for computing $R(D)$ converges because the distortion matrix is over $\mathbb{R}$. Does the same algorithm converge when the distortion matrix has entries in the adele ring $\mathbb{A}_{\mathbb{Q}}$ with componentwise constraints? The convergence proof may need modification for the ultrametric structure.

1. Second-order asymptotics (adelic dispersion). For finite blocklength, the rate-distortion function has a dispersion term. The adelic generalisation would have a vector dispersion — a separate dispersion at each place.

1. Adelic lossy source-channel separation. The standard separation theorem states that a source can be transmitted over a channel iff $R(D) \lt C$. The adelic generalisation would state: transmission is possible iff $R_v(D_v) \lt C_v$ for every place $v$ — a componentwise condition. If the inequality is violated at even one place, the total product capacity is insufficient, regardless of slack at other places.

1. Experimental test for $p$-adic distortion. Is there a physical measurement where the $p$-adic distortion $d_p$ is the operationally correct loss function — i.e., where minimising archimedean error AND $p$-adic error jointly is required for a specific engineering task? This would be the first experimental validation of adelic rate-distortion theory.

#9. Conclusion: The Trilogy Complete

The Adelic Shannon Theory foundation trilogy is now complete:

1. Adelic Shannon Theory [1]: Generalised entropy and channel capacity to the adeles.
2. Adelic Entropic Numbers [2]: Established the entropic number $(x, \mathbf{I}(X))$ as the natural data type.
3. Adelic Rate-Distortion Theory (this paper): Generalised compression and rate-distortion to the adeles.

The three papers form a coherent programme: information in the adele ring is a vector, not a scalar; measurements are entropic numbers, not bare floats; and compression must satisfy componentwise distortion constraints at every place.

The practical consequence is a new class of compression systems — adelic codecs — that preserve both the approximate magnitude AND the divisibility structure of rational data. Whether such codecs can be built, and whether they offer advantages over standard (archimedean-only) compression, are open experimental questions for the next phase of the programme.

#Declarations

Funding: This research received no specific grant from any funding agency.

Conflicts of Interest: The author declares no conflicts of interest.

Author Contributions: Single author.

Data Availability: No experimental data were generated or analysed. Computational examples are reproducible from stated distributions.

Use of Artificial Intelligence: AI-assisted drafting was used for synthesis and prose refinement.

#References

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