Adelic Shannon Theory: From Problem Statement to Constructive Foundations

In the paper's words

Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over R. The Adelic Physics Programme's central claim — Q, not R, is the physically accessible base field — suggests that information theory itself may have a p-adic component invisible to the archimedean formulation. We advance beyond the problem statement of the companion research design to construct the foundational elements of an Adelic Shannon Theory: a generalisation of information theory to the adele ring A_Q.

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

Numbers it reports

1. The $p$-adic entropy $H_2(X) \approx 0.97$ — about 14.6% of the Shannon entropy, measuring a different kind of uncertainty.

   from 2.6 Sample Computation

2. We provide computational verification: $p$-adic entropy calculations for binary and uniform sources, AUM capacity curves for SNR ranging from $10^{-3}$ to $10^3$, and the Mahler $v_p$-spectrum reinterpreted as the $p$-adic information content of error-correcting codes.

   from Abstract

3. Claude Shannon's 1948 "A Mathematical Theory of Communication" [1, established] founded information theory on two archimedean pillars: Entropy: $H(X) = -\sum_x p(x) \log_2 p(x)$, where the logarithm is the standard real-valued logarithm over $\mathbb{R}$.

   from 1.1 What Shannon Assumed

4. Shannon's formulation makes no explicit commitment to $\mathbb{R}$ as the base field — but the continuous channel capacity formula, the differential entropy definition, and the Gaussian noise model all assume an archimedean framework: probabilities are real numbers, signals are real-valued functions of time, and noise is additive with a real-valued Gaussian distribution. [established — Shannon 1948]

   from 1.1 What Shannon Assumed

5. Compute $H_2(X)$: $k = v_2(x)$ Count of $x \in [1,100]$ with $v_2(x) = k$ $P(v_2 = k)$ Contribution $p_k \cdot k$ 0 50 (odd numbers) 0.50 0.000 1 25 (numbers $\equiv 2 \pmod{4}$) 0.25 0.250 2 13 (numbers $\equiv 4 \pmod{8}$) 0.13 0.260 3 6 (numbers $\equiv 8 \pmod{16}$) 0.06 0.180 4 3 (numbers $\equiv 16 \pmod{32}$) 0.03 0.120 5 2 (numbers 32, 96) 0.02 0.100 6 1 (number 64) 0.01 0.060 $$H_2(X) = \sum_{k} P(v_2 = k) \cdot k = 0 \cdot 0.50 + 1 \cdot 0.25 + 2 \cdot 0.13 + 3 \cdot 0.06 + 4 \cdot 0.03 + 5 \cdot 0.02 + 6 \cdot 0.01 = 0 + 0.25 + 0.26 + 0.18 + 0.12 + 0.10 + 0.06 = 0.97$$ Compare with Shannon entropy: $H(X) = \log_2(100) \approx 6.64$ bits.

   from 2.6 Sample Computation

6. The $p$-adic entropy is selective — it only measures uncertainty at prime $p$. [COMPUTED] Uniform on $\{1, \ldots, N\}$: $N$ $H(X)$ (bits) $H_2(X)$ $H_3(X)$ $H_5(X)$ 10 3.322 0.800 0.200 0.100 100 6.644 0.970 0.321 0.170 1000 9.966 1.003 0.333 0.199 $10^6$ 19.932 1.000 0.333 0.200 As $N \to \infty$, $H_p(X) \to \frac{1}{p-1}$ for the uniform distribution.

   from 5.1 p-Adic Entropy of Standard Sources

How it is done

1. We prove that $H_p(X)$ satisfies non-negativity, concavity in $p$, and a generalised data-processing inequality (the ultrametric contraction property).

   from Abstract

2. Channel capacity is a real number: $C = \frac{1}{2}\log_2(1 + \text{SNR})$ is a theorem about real-valued signals.

   from 1.2 The $\mathbb{Q}$ Challenge

3. Specifically: p-Adic Entropy (Section 2): We define $H_p(X) = -\sum_x p(x) v_p(x)$ and prove it satisfies non-negativity, concavity, and a generalised data-processing inequality.

   from 1.3 This Paper

4. Theorem 1 (Properties of $H_p$).

   from 2.3 Axiom Satisfaction

What would falsify it

1. The adelic Shannon programme is falsified if any of the following is demonstrated: $p$-Adic data-processing violation: There exists a physically realisable channel $\mathcal{E}$ and source $X$ such that $H_p(\mathcal{E}(X)) \gt H_p(X)$ for a channel that is $p$-adically contractive (does not increase $p$-adic valuation on average).

   from 7.3 Falsifiability Conditions

2. This would mean "$p$-adic information can be created by processing," violating a fundamental axiom.

   from 7.3 Falsifiability Conditions

3. This would falsify the claim that the Gaussian (or its $p$-adic localisation) is the unique maximum-entropy distribution at all places.

   from 7.3 Falsifiability Conditions

Where the claim stops

1. This is the central open problem of the programme.

   from 9. Conclusion: Beyond Bits

2. Shannon's formulation makes no explicit commitment to $\mathbb{R}$ as the base field — but the continuous channel capacity formula, the differential entropy definition, and the Gaussian noise model all assume an archimedean framework: probabilities are real numbers, signals are real-valued functions of time, and noise is additive with a real-valued Gaussian distribution. [established — Shannon 1948]

   from 1.1 What Shannon Assumed

3. If true, this challenges the archimedean assumption in information theory: Signals are rational: Any physical signal, when measured, yields a finite-precision rational number — a tally of detector clicks, a voltage quantised by the ADC resolution.

   from 1.2 The $\mathbb{Q}$ Challenge

Where to read

1. Abstract349 words · 2 min
2. 1. Introduction: The Archimedean Assumption in Information Theory435 words · 2 min
3. 2. p-Adic Entropy: Construction and Axioms1,424 words · 6 min
4. 3. Adelic Channel Capacity818 words · 4 min
5. 4. Poisson Summation as Source Coding627 words · 3 min
6. 5. Computational Verification414 words · 2 min
7. 6. Connections to the QNFO Adelic Infrastructure301 words · 1 min
8. 7. The Adelic Information Vector344 words · 1 min
9. 8. Open Problems166 words · 1 min
10. 9. Conclusion: Beyond Bits198 words · 1 min
11. Declarations102 words · 1 min