Villaber Universe Formula Collection: The Invariant Sum (1/2)^n = 1, Classical Collatz 3x+1, kx+1 Family, and Riemann Critical Line - A 10-Poster Visual Synthesis (2026)

Villaber Universe Formula Collection - A 10-Poster Visual Synthesis by Ruel Villaber, Mandaluyong City, Philippines (2026). This collection presents a unified educational synthesis from an elementary invariant to two major unsolved problems. PART I: THE FUNDAMENTAL INVARIANT (Posters 1-4) Foundation is the convergent geometric series sum_{n=1}^{inf} (1/2)^n = 1. Equivalent forms: 1 = 1/2 + 1/2, S_n = 1 - (1/2)^n -> 1, and binary proof 0.111..._2 = 1 (2x = 1 + x => x = 1). Includes Tree Diagram of dyadic fractions: 1 = 4 * 1/4 = 8 * 1/8 = 16 * 1/16, and the invariant statement 0.5 + 0.5 = 1.0. PART II: CLASSICAL COLLATZ (Posters 5-6) Poster: Classical Collatz 3x+1 Definition: C(n) = n/2 if even, 3n+1 if odd. Known conjecture: All paths lead to the 4-2-1 loop. Example n=7 verified: 7->22->11->34->17->52->26->13->40->20->10->5->16->8->4->2->1. Verified up to 2^68, general proof remains open. Poster: kx+1 Family Extension - Why k=3 is Special. C_k(n) = n/2 if even, k*n+1 if odd. k=1 Trivial convergence. k=3 Classical conjectured 4-2-1 loop. k=5 Counter-example loop: n=13 -> 66->33->166->83->416->208->104->52->26->13 (13-loop). k=7 Divergent: n=3 -> 22->11->78->39->274->... increasing. Conclusion: Not all k converge to 1, only k=3 is the classical case. PART III: RIEMANN ZETA (Posters 7-9) Definition: zeta(s) = sum_{n=1}^{inf} 1/n^s for Re(s) > 1, with analytic continuation via functional equation. Critical Strip: 0 < Re(s) < 1. Critical Line: Re(s) = 0.5. Numerical verification of first three non-trivial zeros: s = 0.5 + 14.1347i (zeta ~ 2.7e-11), s = 0.5 + 21.0220i (zeta ~ 8.1e-11), s = 0.5 + 25.0108i (zeta ~ 0). All zeros lie on Critical Line. PART IV: UNIFICATION (Poster 10) The Millennium Prize & The Villaber Invariant collage. The invariant sum (1/2)^n = 1 presented as inspirational framework for 7 Clay Problems: Riemann (Re=0.5), P vs NP (Kraft=1), Navier-Stokes (Vol=1), Yang-Mills (Gap=1), BSD (L=1), Hodge (Hodge=1), Poincare (S^3=1 Solved by Perelman). This is a philosophical analogy / pattern, not a formal claimed solution to the 6 remaining unsolved problems. Author: Ruel Villaber Year: 2026 Collection: 10 Posters Purpose: Educational, visual learning, independent research framework. License: CC BY 4.0