What this page contains

This page collects the public research outputs of the MAX Prime Theory: peer-style preprints (Zenodo) and reproducible public demos (Hugging Face). The MAX Prime Theory introduces an index pre-sieve and a CRT-based modular partition that allow one to organize prime candidates into classes with constant modular signatures.

Note: the MAX App is the practical application layer that uses these ideas for identity/login. The MX² container (used to package secrets safely) has its own dedicated page: MX² (MX2).

MAX ID (high-level idea)

The MAX ID can be described as a mathematical identity derived from a deterministic process that produces structured prime candidates and (under standard heuristic assumptions such as Bateman–Horn) predicts infinitely many primes in admissible classes. The identity concept is built around modular signatures induced by filters on the indices of polynomial sequences (rather than filtering prime values a posteriori).

Concretely, the research focuses on affine-related quadratic polynomials and shows that once an index class is fixed (via congruence conditions modulo a finite set of primes), the values produced by that class carry a constant modular signature modulo each prime in the chosen set. This gives a constructive way to define and explore prime-candidate classes by signature.

Papers (Zenodo • MAX Prime Theory Community)

The three papers below are part of the MAX Prime Theory series. Paper 1 introduces the index pre-sieve and experimental evidence on density/enrichment. Paper 2 develops the induced modular partition and constant signatures (CRT structure). Paper 3 extends the framework to a parametric family.

• Zenodo • Paper 1 — Affine-related quadratic polynomials, an index pre-sieve and finite truncations of Bateman–Horn local factors: an experimental study
• Zenodo • Paper 2 — A uniform modular partition of prime candidates for a pair of affine-correlated quadratic polynomials based on an index pre-sieve
• Zenodo • Paper 3 — A parametric family of affine-related quadratic polynomials: index pre-sieve, modular partitions, and Bateman–Horn classes

Public demos (Hugging Face)

These interactive demos expose sequences with and without modular filters and allow independent inspection of the induced modular signatures.

• Hugging Face • Sequence-A (no filters)
• Hugging Face • Sequence-A (mod 3)
• Hugging Face • Sequence-A (mod 7)
• Hugging Face • Sequence-A (mod 3+7)
• Hugging Face • Sequence-A (3+37, esc=5)
• Hugging Face • MAX-Test

MAX App (practical access)

The MAX App uses this research as a foundation to provide a practical flow for identity/login and to let users explore modular signatures derived from their own deterministic setup. This website page is intentionally focused on the public research layer (papers + demos), while application details live in the MAX App sections.

To view prime families organized by modular signature in the reserved area, log in using the MAX App (“Login with MAX”) and open the Modular Signatures section.

Contact: max@max-russo.com • LinkedIn