When Will Non-Archimedean Geometry Displace the Real Numbers? A Structured Assessment of the Adelic Substrate Thesis

In the paper's words

The real numbers are the default continuum for physical modeling — every quantum state lives in a Hilbert space over C, every neural network optimizes over R^n, every error-correcting code is designed with Euclidean distance. Yet Ostrowski's theorem (1916) demonstrates that the real numbers are merely one of many completions of the rationals: every non-trivial absolute value on Q is equivalent to either the Archimedean absolute value or a p-adic absolute value for some prime p. The Archimedean default is a choice, not a necessity.

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

Numbers it reports

1. Most importantly, a $p$-adic architecture must demonstrate a benchmark improvement over Euclidean baselines on at least one standard task — a 3–5% improvement in top-1 accuracy would constitute a compelling existence proof.

   from 3.2 Candidate B: $p$-adic Neural Architectures (Machine Learning)

2. Based on the qualitative ranking and timeline constraints, we recommend the following allocation of research effort: Rank Candidate Allocation Rationale A Ultrametric QEC codes 35% Shortest timeline, highest testability, clear success metric (code threshold).

   from 7 Research Effort Allocation

3. Hedge Unknown candidates 10% Anti-fragility floor — reserve effort for ideas not yet conceived.

   from 7 Research Effort Allocation

4. At least one peer-reviewed publication demonstrates a quantum error-correcting code whose construction is native to a Bruhat–Tits tree (with ultrametric codeword distance), and the code's threshold is within 20% of the surface-code threshold under identical noise models.

   from 8 Calibration Register

5. An ultrametric QEC code is demonstrated in simulation with a threshold exceeding 0.5% under circuit-level depolarizing noise — within a factor of two of the surface-code threshold of approximately 1%.

   from 8 Calibration Register

6. Archimedean quantum mechanics has 100 years of experimental validation with zero anomalies.

   from 3.3 Candidate C: Non-Archimedean Quantum Foundations

How it is done

1. Yet Ostrowski's theorem (1916) demonstrates that the real numbers are merely one of many completions of the rationals: every non-trivial absolute value on $\mathbb{Q}$ is equivalent to either the Archimedean absolute value or a $p$-adic absolute value for some prime $p$.

   from Abstract

2. Ostrowski's theorem establishes that every non-trivial absolute value on $\mathbb{Q}$ is equivalent to either the usual real absolute value $|\cdot|_\infty$ or the $p$-adic absolute value $|\cdot|_p$ for some prime $p$.

   from 1 Introduction

3. Most importantly, a $p$-adic architecture must demonstrate a benchmark improvement over Euclidean baselines on at least one standard task — a 3–5% improvement in top-1 accuracy would constitute a compelling existence proof.

   from 3.2 Candidate B: $p$-adic Neural Architectures (Machine Learning)

4. The most actionable approach is a theoretical reconnaissance: build the $p$-adic QM formalism, identify a testable signature (a Bell-type experiment where $p$-adic and standard QM predict different correlations), and wait for experimentalists to reach the required precision.

   from 3.3 Candidate C: Non-Archimedean Quantum Foundations

What would falsify it

1. The null hypothesis — that non-Archimedean modeling remains a niche mathematical tool — is the well-anchored prediction.

   from Abstract

2. This paper provides a structured assessment of that question, identifying the most plausible candidate domains, anchoring probability judgments to historical reference classes, and registering dated, falsifiable predictions.

   from 1 Introduction

3. The most actionable approach is a theoretical reconnaissance: build the $p$-adic QM formalism, identify a testable signature (a Bell-type experiment where $p$-adic and standard QM predict different correlations), and wait for experimentalists to reach the required precision.

   from 3.3 Candidate C: Non-Archimedean Quantum Foundations

Where the claim stops

1. We find that the central bet — displacement in at least one domain by 2036 — has a calibrated probability of approximately 0.15–0.20, consistent with a reference class in which zero of the three closest historical analogues (non-Euclidean geometry → general relativity, abstract algebra → quantum mechanics, information theory → biology) achieved mainstream adoption within a decade.

   from Abstract

2. The molecular clock hypothesis [1] assumes constant mutation rates across lineages, producing ultrametric distance matrices.

   from 1.1 The Existence Proofs

3. Current surface codes achieve thresholds of approximately $1\%$ under circuit-level depolarizing noise using Euclidean sphere-packing geometry.

   from 3.1 Candidate A: Ultrametric Quantum Error-Correcting Codes (QEC)

Where to read

1. Abstract240 words · 1 min
2. 1 Introduction484 words · 2 min
3. 2 Domain Assessment162 words · 1 min
4. 3 Candidate Assessment839 words · 4 min
5. 4 Historical Reference Class and Calibration309 words · 1 min
6. 5 Adversarial Assessment419 words · 2 min
7. 6 Judgment Sensitivity158 words · 1 min
8. 7 Research Effort Allocation183 words · 1 min
9. 8 Calibration Register401 words · 2 min
10. 9 Practical Applications162 words · 1 min
11. 10 Counterfactual Backcasting328 words · 1 min
12. 11 Conclusion239 words · 1 min