DichotomY IdentitY: Euler-Bernoulli Numbers, Sets-Multisets, FD-BE Quantum-Statistics, 1/f$\backslash $\^{}{\{}0{\}}-1/f$\backslash $\^{}{\{}1{\}} Power-Spectra, Ellipse-Hyperbola Conic-Sections, Local-Global Extent: "Category-Semantics"

Seminal Apostol[Math.Mag.81,3,178(08);Am.Math.Month.115,9,795(08)]-Rota[Intro.Prob. Thy.(95)-p.50-55] DichotomY equivalence-class: set-theory: sets V multisets; closed V open; to Abromowitz-Stegun[Hdbk.Math.Fns.(64)]-ch.23,p.803!]: numbers/polynomials generating-functions: Euler V Bernoulli; to Siegel[Schrodinger Cent.Symp.(87); Symp.Fractals, MRS Fall Mtg.,(1989)-5-papers!] power-spectrum: 1/f$\backslash $\^{}{\{}0{\}}-White V 1/f$\backslash $\^{}{\{}1{\}}-Zipf/Pink (Archimedes) HYPERBOLICITY INEVITABILITY; to analytic-geometry Conic-Sections: Ellipse V (via Parabola) V Hyperbola; to Extent/Scale/Radius: Locality V Globality, Root-Causes/Ultimate-Origins: Dimensionality: odd-Z V (via fractal) V even-Z, to Symmetries/(Noether's-theorem connected)/Conservation-Laws Dichotomy: restored/conservation/convergence=0- V broken/non-conservation/divergence=/=0: with asymptotic-limit antipodes morphisms/ crossovers: Eureka!!!; "FUZZYICS"=''CATEGORYICS''!!! Connection to Kummer(1850) Bernoulli-numbers proof of FLT is via Siegel(CCNY;1964) $<$ (1994)[AMS Joint Mtg. (2002)-Abs.973-60-124] short succinct physics proof: FLT = Least-Action Principle!!!