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Bibliographic Details
Main Author:	Fedor, Kapitanov
Format:	Recurso digital
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Published:	Zenodo 2026
Online Access:	https://doi.org/10.5281/zenodo.19019202
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Table of Contents:

We establish an algebraic stability criterion for particle-like structures on a discrete carrier, derived from the closure condition of iterated distinction and the Euler identity. The closure condition x² + x − 1 = 0 and the Euler identity eiπ + 1 = 0 unite in a single equation with no free parameters: x² + x + eiπ = 0 We prove this unification is not a tautology. The two components were derived by independent paths separated by two centuries of mathematics. The closure condition arises from the algebra of iterated distinction with no geometric input. The Euler identity arises from complex analysis with no reference to distinction. Their meeting at −1 is a structural fact, not a definition. We prove the Dimensional Guard Theorem: the irrational numbers φ, π, and e prevent dimensional collapse at distinct levels of the source–carrier hierarchy: <ul> <li>φ guards the source algebra: the distinction gap g(x) = x(1−x) > 0 is never rationally closed</li> <li>π guards the 2D carrier geometry: the geometric differential δ = π − 3 > 0 is always non-zero</li> <li>e guards the 3D carrier flow: the aperture gap 2πe − 4φ³ > 0 is always non-zero</li> </ul> Their irrationality is a structural necessity, not an accident. Mass is not a primitive property of matter. It is the geometric cost of topological self-closure on the carrier. The free propagation state has zero closure cost and zero mass. Mass emerges from the act of closure itself, not from interaction with an external field. The minimum non-zero closure cost is set by the geometric differential δ = π − 3 — the irreducible gap between the continuous circle and its FCC-compatible 12-gon realization with coordination number k = 12. No stable massive structure can have a closure cost smaller than this minimum. We establish the Sieve Invariance Principle: the mass ratio mτ/me admits two independent representations: <ul> <li>Static (geometric): π⁴ / (π − 3) × D ≈ 3440</li> <li>Dynamic (operator cascade): 4φ³ × k × 2πe ≈ 3473</li> </ul> Experimental value: 3477.2. The difference (0.96%) is the structural aperture gap of the carrier — the tolerance required to prevent singular rigidity of the lattice. Without this tolerance, the carrier would be infinitely rigid and could not support wave propagation. Central result: Mass is not a primitive property of matter. It is the geometric cost of topological self-closure on the carrier, governed by the stability criterion: x² + x + eiπ = 0 Keywords: mass generation, closure condition, Euler identity, stability criterion, geometric differential, golden ratio, FCC lattice, dimensional guard, sieve invariance, ontological resolution theory, ORT

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