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

In the paper's words

The Adelic Shannon Theory [1] generalised channel capacity to the adele ring A_Q. The companion bridge paper [2] established that the entropic number $(x, S)$ from Measurement Stratigraphy [3] is precisely the adelic information vector I(X) = (I_∞, I₂, I₃, ldots). This paper completes the trilogy by generalising rate-distortion theory to the adelic setting.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. 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.

   from 7.1 SRE as Adelic Rate-Distortion

How it is done

1. We define the $p$-adic distortion measure $d_p(x, \hat{x}) = p^{-v_p(x - \hat{x})}$ — the natural ultrametric distortion derived from the $p$-adic absolute value — and derive the adelic rate-distortion function $R(D_\infty, D_2, D_3, \ldots)$: the minimum rate (in adelic bits) required to represent a source with distortion at most $D_v$ at each place $v$.

   from Abstract

2. For a Gaussian source with squared-error distortion: $$R(D) = \frac{1}{2}\log_2\left(\frac{\sigma^2}{D}\right), \quad 0 \leq D \leq \sigma^2$$ 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 $\mathbb{R}$.

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

3. Section 4 proves the Shannon lower bound.

   from 1.3 Structure

4. Theorem 1 (Factorisable Source Rate-Distortion).

   from 3.2 Factorisable Sources

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. 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.

   from 9. Conclusion: The Trilogy Complete

2. For a Gaussian source with squared-error distortion: $$R(D) = \frac{1}{2}\log_2\left(\frac{\sigma^2}{D}\right), \quad 0 \leq D \leq \sigma^2$$ 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 $\mathbb{R}$.

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

3. 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).

   from 1.2 The Naturalness of $p$-Adic Distortion

Where to read

1. Abstract287 words · 1 min
2. 1. Introduction: The Third Pillar423 words · 2 min
3. 2. The Adelic Distortion Measure324 words · 1 min
4. 3. The Adelic Rate-Distortion Function427 words · 2 min
5. 4. The Adelic Shannon Lower Bound279 words · 1 min
6. 5. The Hardest Source: Gaussian Entropic Number282 words · 1 min
7. 6. Computational Verification328 words · 1 min
8. 7. Connections to QNFO Infrastructure204 words · 1 min
9. 8. Open Problems225 words · 1 min
10. 9. Conclusion: The Trilogy Complete136 words · 1 min
11. Declarations54 words · 1 min