Universal Anderson–Faulhaber–Bernoulli Identity: Internal Structure of Perfect Powers and Arithmetic Obstruction via Discrete Calculus

This paper presents, in a unified form and with explicit logical justification, a chain of original results on the internal structure of perfect powers and their connection with Fermat’s Last Theorem (FLT). Starting from the historical formula of Nicomachus of Gerasa (c. 100 AD) for the cumulative sum of cubes, S_3 (n)=T_n^2, and applying the backward finite-difference operator ∇ —formalised by Taylor (1715) and systematised by Boole (1860)— the Anderson Identity (2026) is derived: n^3=n^2/4 [(n+1)^2-(n-1)^2 ]=T_n^2-T_(n-1)^2. This identity, valid exclusively for cubes, is extended to every p≥1 through the Universal Anderson–Faulhaber–Bernoulli Identity (2026): n^p=1/(p+1) ∑_(j=0)^p((p+1)¦j) B_j^+ δ_(p+1-j) (n), δ_m (n) (≔) n^m-(n-1)^m, derived by applying the operator ∇ to the classical Faulhaber–Bernoulli formula for cumulative sums of powers. The quantity δ_m (n)=n^m-(n-1)^m —the finite difference of the individual m-th power— constitutes the original internal perspective of this work: it reorients the historical cumulative-sum formula toward individual powers, revealing that the internal algebraic complexity of n^p grows as C(p)=⌊p/2⌋+1, with p=3 as the unique point of optimal compactness (pure monomial). The second contribution is the Universal Symbolic Representation (2026): h=√(p&I_p (a)+I_p (b) ), h∉Z ∀ p≥3, which expresses h=√(p&a^p+b^p ) through purely integer operations in the radicand and establishes that its irrationality for p≥3 is a structurally inevitable consequence, not an accidental one. Full step-by-step derivations, explicit expansions for p=2,…,8, 50-digit-precision numerical verifications for 10 000 pairs (a,b) with 1≤a≤b≤100, and the conceptual gradation of the Fermatian obstruction in three regimes —quadratic, cubic, and Bernoulli— are presented. The Structural Stratification Theorem is proved: C(p)=⌊p/2⌋+1, with p=3 as the unique point of optimal compactness. Complete chronological historical contextualisation from the Pythagoreans to Wiles, analysis of the originality of the present perspective, generalised symmetry breaking, and genuine pedagogical value are included.