Number-Theoretic Ultrametric Foundations: A Unified p-adic Framework for Error-Correcting Code Classification

In the paper's words

A unified framework connecting p-adic valuation theory, Mahler spectral expansions, Kodaira-Neron fiber classification, and the Amice transform to the classification of quantum error-correcting codes. Three conjectures with 14 lemmas, computational verification across 4 code families at 83% classification rate.

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

Numbers it reports

1. Results: 83% classification rate across 50 test codes from all four families 100% lemma-level agreement on correctly classified codes Surface codes → $I_4^*$ type (92% of cases) CSS codes → $I_n$ type (78% of cases, $n$ varies with distance) Optimal codes → $II^*$ or $III^*$ (91% of cases) Random codes → $I_1$ type (96% of cases — generic smooth reduction)

   from 4.4 Computational Prototype

2. The observed 83% experimental rate is consistent with this bound for moderate $n$. $\square$

   from Lemma 10: Classification Rate Bound

3. The Kodaira-Néron classifier achieves 83% classification accuracy with 100% lemma-level agreement; the Mahler spectral analysis yields $v_p^{\max}=28$ for optimal codes versus $v_p^{\max}=4$ for random ensembles.

   from Abstract

4. Computational verification achieves 83% classification rate with 100% lemma-level agreement.

   from 1.3 Principal Results

5. Test Suite Results (Unified v1.0): Metric Value Total test codes 200 Successful classifications 166 (83.0%) Lemma-level agreement 100% (14/14 lemmas verified) C7.3' spectral gap verification 187/200 (93.5%) C2.1' tree consistency 151/200 (75.5%) C5.1 Kodaira-Néron match 166/200 (83.0%)

   from 8.2 Unified Test Suite

6. All tests were run on a standard workstation (AMD Ryzen 9, 32GB RAM, Windows 11).

   from 8.3 Hardware and Performance

How it is done

1. We provide formal proof sketches across 14 constituent lemmas (5+5+4 split) and report computational verification across 4 code families.

   from Abstract

2. Computational verification achieves 83% classification rate with 100% lemma-level agreement.

   from 1.3 Principal Results

3. The paper is organized as follows. §2–§8 develop each of the seven pillars with formal definitions, key theorems, and computational prototypes. §9 formulates the three principal conjectures. §10–§12 present the proof sketch summaries (5+5+4 lemmas). §13 reports computational results. §14 discusses implications and future work.

   from 1.4 Organization

4. This theorem justifies the central role of $p$-adic methods: for any ultrametric structure arising naturally (as in code trees), the associated completion must be a $p$-adic field for some prime $p$.

   from 2.3 Ostrowski's Theorem and Classification

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. Future work will focus on formalizing the 14 lemmas into full proofs, extending to $p$-ary codes, and developing the constructive applications for optimal code design based on Kodaira-Néron fiber engineering.

   from 15. Conclusion

2. Conjecture Status Matrix (Final): Code Family C7.3' C2.1' C5.1 Aggregate Surface Codes PASS PASS FAIL 2/3 CSS Codes WEAK WEAK PASS 1/3 Optimal Codes PASS WEAK PASS 3/3 ★ Random Codes PASS WEAK PASS 3/3 ★ ★ Optimal and Random codes satisfy all three conjectures, establishing them as theorem targets for formal proof development.

   from 1.3 Principal Results

3. The paper is organized as follows. §2–§8 develop each of the seven pillars with formal definitions, key theorems, and computational prototypes. §9 formulates the three principal conjectures. §10–§12 present the proof sketch summaries (5+5+4 lemmas). §13 reports computational results. §14 discusses implications and future work.

   from 1.4 Organization

Where to read

1. A Unified p-adic Framework for Error-Correcting Code Classification24 words · 1 min
2. Abstract172 words · 1 min
3. 1. Introduction396 words · 2 min
4. 2. Pillar I: $p$-adic Valuation Theory340 words · 1 min
5. 3. Pillar II: Mahler Spectral Expansions286 words · 1 min
6. 4. Pillar III: Kodaira-Néron Classification342 words · 1 min
7. 5. Pillar IV: Amice Transform and $p$-adic Fourier Theory220 words · 1 min
8. 6. Pillar V: Error-Correcting Codes as Ultrametric Trees274 words · 1 min
9. 7. Pillar VI: Bruhat-Tits Buildings215 words · 1 min
10. 8. Pillar VII: Computational Verification174 words · 1 min
11. 9. Principal Conjectures305 words · 1 min
12. 10. Proof Sketch I: Spectral Bounds (Lemmas 1–5)420 words · 2 min