Gorde:
| Egile nagusia: | |
|---|---|
| Formatua: | Recurso digital |
| Hizkuntza: | ingelesa |
| Argitaratua: |
Zenodo
2025
|
| Gaiak: | |
| Sarrera elektronikoa: | https://doi.org/10.5281/zenodo.17728408 |
| Etiketak: |
Etiketa erantsi
Etiketarik gabe, Izan zaitez lehena erregistro honi etiketa jartzen!
|
Aurkibidea:
- <p>This paper is part of the Axiom Zero (AZ) series and assumes familiarity with the main AZ framework (survivors, AZ-primes, horizon windows, and the structural laws collected in the “Laws, Principles, and Rules” note). New readers should begin with the foundational manuscript <em>Axiom Zero: Structural Irreducibility and the Unpredictability of Primes</em> before using this paper as a structural case study.</p> <p>Working entirely inside AZ’s arithmetic layer, this note gives an internal proof of Lagrange’s four-square theorem: every natural number is a sum of four squares. It constructs an “AZ quaternion ring” QAZ≅Z4Q_{\text{AZ}} \cong \mathbb{Z}^4QAZ≅Z4, defines a norm<br>N(a,b,c,d)=a2+b2+c2+d2N(a,b,c,d) = a^2 + b^2 + c^2 + d^2N(a,b,c,d)=a2+b2+c2+d2, proves that this norm is multiplicative, and then shows that every AZ-prime arises as such a norm using only Peano arithmetic, finite combinatorics modulo an AZ-prime, and a descent argument. By conservativity of AZ over (N;0,1,+,⋅,≤)(\mathbb{N};0,1,+,\cdot,\le)(N;0,1,+,⋅,≤), the internal AZ theorem immediately yields the classical Lagrange four-square theorem in the standard model of the natural numbers.</p> <p>For orientation and cross-references, readers may wish to consult the companion AZ preprints on Zenodo:</p> <ul> <li> <p><em>Axiom Zero: Structural Irreducibility and the Unpredictability of Primes </em>— DOI: <a href="https://doi.org/10.5281/zenodo.16998285">10.5281/zenodo.16998285</a></p> </li> <li><em>Axiom Zero: Laws, Principles, and Rules of Structure </em>— DOI: <a href="https://doi.org/10.5281/zenodo.17728204">10.5281/zenodo.17728204</a></li> <li> <p><em>Deck Limit Laws in Axiom Zero</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17728647">10.5281/zenodo.17728647</a></p> </li> <li><em>No Third Mechanism in Axiom Zero: Structural Completeness and Non–Existence Schemes</em> — DOI: <a href="https://doi.org/10.5281/zenodo.18097942">10.5281/zenodo.18097942</a></li> <li> <p><em>Dyadic Rigidity and the Non-Existence of Odd Perfect Numbers in Axiom Zero</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17886421">10.5281/zenodo.17886421</a></p> </li> <li><em>Channel Non-Extinction in Axiom Zero </em>— DOI: <a href="https://doi.org/10.5281/zenodo.18099516">10.5281/zenodo.18099516</a></li> <li> <p>Twin Primes in Axiom Zero: Horizon Contribution, Uniformity, and No Third Mechanism — DOI: <a href="https://doi.org/10.5281/zenodo.17903043">10.5281/zenodo.17903043</a></p> </li> <li> <p><em>Goldbach in Axiom Zero: A Deterministic Additive-Sieve </em>— DOI: <a href="https://doi.org/10.5281/zenodo.17728745">10.5281/zenodo.17728745</a></p> </li> <li> <p><em>Conservativity of Axiom Zero over N</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17073211">10.5281/zenodo.17073211</a></p> </li> <li> <p><em>Analytic Conservativity of Axiom Zero</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17073257">10.5281/zenodo.17073257</a></p> </li> <li> <p><em>Stress Testing Axiom Zero: Computational Falsification Attempts and Classical Consistency Checks</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17728602">10.5281/zenodo.17728602</a></p> </li> <li> <p><em>Contribution Classes and Horizon Contribution</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17110018">10.5281/zenodo.17110018</a></p> </li> <li> <p><em>Euclid–Euler in Axiom Zero: Even Perfect Numbers from Structural Arithmetic</em> — DOI: <a href="https://doi.org/10.5281/zenodo.17728314">10.5281/zenodo.17728314</a></p> </li> </ul> <p>Together, these works position Axiom Zero as a conservative, structural complement to classical number theory—rebuilding Euclid–Euler and Lagrange internally before pushing toward deeper additive and correlation problems.</p> <p>For questions or comments, contact: axiomzero.math@gmail.com</p>