P/PN — Part I of II: Exterior Arithmetic, Primitive Bivectors, and Computational Verification Certificates

Public Mathematical Research Package Researcher: Philippe Beauchamp ORCID: 0009-0003-7407-394X Repository: jackophil-dev/public-math-research Release: P/PN — Part I of II: Exterior Arithmetic, Primitive Bivectors, and Computational Verification Certificates Overview This deposit collects a structured set of verified mathematical calculations, algebraic certificates, and foundational structural statements concerning exterior arithmetic lattices, primitive bivectors, and invariant constraints, developed under a strict zero-assumption protocol (Règle Zéro). The material is organized into a rigorously verified exterior arithmetic core (Bloc A) while explicitly isolating the subsequent geometric identification problem (Bloc B). Each calculation and certificate is structured to be independently inspectable, reproducible, and verifiable. Speculative data reconstructions and forced embeddings are strictly forbidden. Research Structure & Scope (Part I of II) This release represents Part I of the P/PN research program, focusing exclusively on established arithmetic structures and computational verifications: Ambient Exterior Lattice ($\Lambda^2 L_6$): Formal definition of the free $\mathbb{Z}$-module of rank 6 and its canonical tensor/exterior projections $\pi(x \otimes y) = x \wedge y$. Explicit Source Vectors & Relations: Controlled initialization with verified source vectors satisfying internal relations ($a = b$, $c = 2d - a$). Primitive Bivector Construction ($v = d \wedge b$): Exact coordinate expansion across the 15-dimensional standard bivector basis. Smith Normal Form & Saturation: Rigorous $\gcd=1$ verification proving strict primitivity and pure rank-1 direct summand structure ($\operatorname{SNF}(v) = \operatorname{diag}(1, 0, \ldots, 0)$). Provenance Audit & Gate Status: Formal locking of Bloc A (FROZEN) and documentation of the open status of the geometric identification gate for $M_\iota$ (Bloc B — Source Not Found). Academic Abstract This research package presents the verified foundational exterior arithmetic layer of the P/PN research program. Operating under a strict non-speculative paradigm (Règle Zéro), the work establishes the algebraic integrity of exterior square lattices over $L_6 \simeq \mathbb{Z}^6$, computes canonical integral projections, and proves the strict primitivity of key bivector generators via explicit coordinate expansion and Smith Normal Form (SNF) analysis. This deposit deliberately separates established arithmetic results from the remaining geometric realization problem. The arithmetic and computational components documented here are presented as verified, immutable results; the remaining connection with the intended cohomological realization into $H^1(E^4, \mathbb{Z})$ via an immersion matrix $M_\iota$ is explicitly identified as an open research step (Part II). Scientific Status & Notice to the Mathematical Community The Current State: The public repository and this Zenodo deposit provide verifiable structural milestones, intermediate calculations, explicit lattice formulations, and computational certificates concerning exterior arithmetic and primitive generators. Règle Zéro Compliance: Absolute prohibition of unverified data reconstructions or speculative matrices. What is archived is mathematically proven and computationally certified; what is missing is formally declared as open. Attribution Policy: Released under CC BY 4.0 © 2026 Philippe Beauchamp. Any institution, research group, or automated synthesis pipeline making substantive use of these frameworks is expected to preserve full academic attribution to the author. "Author's Note: This research package establishes a pristine, verifiable arithmetic core under strict zero-assumption protocols. All bivector expansions, SNF calculations, and projection rules are fully open for independent verification. Part II will follow upon the completion of authenticated geometric constraints."