On the Various Formulations of Entropy and the Derivation of Quantum Mechanics

This paper introduces indistinctio per gradus (IPG) — the condition of non-distinctionthroughout the grades of a tolerance filtration — as a new and primitive convergence definitionthat subsumes all classical analytic convergence notions. Metric convergence, topologicalconvergence, R-limits, Cauchy sequences, and C-convergence are all special cases of IPG, obtained by imposing additional structure that the condition itself does not require. IPG isstated as a theorem — every analytic convergence notion is downstream of the bare conditionof non-distinction — and then converted to a definition, internalizing the theorem’s content. א0The Tree of Continua Ck =A^A (A uncountable, |A| = 2) is informationally complete: itcontains all possible sequences, hence all real numbers, hence all eigenvalues of all densitymatrices. The periodic orbits are dense in C. IPG is the convergence structure this densityforces. As a first major application, IPG gives the first combinatorial proof of the Frank–Liebinequality 10 for all α ≥ 1: the α = 2 case is four lines (D2 = 12 ∥μ, Q∥2HS ≥ 0, a Hilbert–Schmidt norm squared) ; the general case follows from informational completeness and theIPG condition at ∞ ∈ S = N ∪ ∞. No complex interpolation. No three-lines theorem. No calculus. As a second application, IPG unifies all entropy families. Shannon, Rényi, Tsallis, vonNeumann, sandwiched Rényi, and entanglement entropy are all readings of a single three-parameter object: Z (α, χ, ε) = Zsym (α) + χ · Zanti (α) + ε · Zβ0 (α), where α ∈ 0, ∞) is the Rényi order parameter, χ ∈ 0, 1 distinguishes classical fromquantum, and ε measures the cohomological depth of entanglement. Shannon and vonNeumann entropy are IPG readings at the limit point [∞ ∈ S, witnessed by single periodicorbits. Rényi and Tsallis entropy are exact readings at finite grades. No analytic limit isrequired in the definition. be quadratic. pi = |ψi |2 is the unique minimal U (1) -invariant probability measure on theprojective space CPn−1. Not chosen. Forced. Every postulate of quantum mechanics is now a theorem: