Rigorous justification of finite residual values in divergent physical sums (Casimir effect)

Establish a rigorous mathematical justification for the appearance of the finite residual value −1/12 in the Casimir effect and related quantum field theory contexts involving divergent sums, ideally by formalizing the cancellation of infinite parts or constructing a precise smoothing within a hyperreal summation/integration framework.

Background

The paper discusses how divergent sums like 1+2+3+… can, via techniques such as zeta regularization, be associated with finite residual values (e.g., −1/12), which have empirical support in physical phenomena like the Casimir effect.

While hyperreal summation and integration may offer a pathway to make this precise by separating infinite and finite parts, the author notes the current absence of a rigorous derivation connecting these values to the physical results.