Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

===== If you are committed to the approach using the extended linear limit LLL and total multiplication, then you must explicitly add and prove the additional properties that bridge the gap to classical functions. Concretely you must show: =====
# (C1) Classical-closure: the subspace C\mathcal{C}C of classical real-valued C∞C^\inftyC∞ functions is closed under the topology/operation in which you apply LLL. Equivalently, whenever a sequence of classical functions {fn}\{f_n\}{fn} is such that Ln→∞fnL_{n\to\infty} f_nLn→∞fn exists, then Ln→∞fnL_{n\to\infty} f_nLn→∞fn is a classical function (and coincides with the classical pointwise / uniform limit when that exists).
# (C2) Compatibility with nonlinearities: for sequences fn,gnf_n,g_nfn,gn of classical functions, L(fngn)=L(fn)⋅L(gn)L(f_n g_n) = L(f_n)\cdot L(g_n)L(fngn)=L(fn)⋅L(gn) whenever products make sense classically — and the same commuting property for derivatives and integrals (i.e., LLL commutes with differentiation and integration on the relevant sequences).
# (C3) Continuity properties: LLL preserves norm bounds needed for energy estimates: for example, if ∥fn∥Hs\|f_n\|_{H^s}∥fn∥Hs is uniformly bounded then L(fn)L(f_n)L(fn) lies in HsH^sHs and the same norm estimates transfer.

If you can construct such an LLL (or prove these properties for a known generalized function algebra), then the Picard-limit argument becomes legitimate and the limit will be classical. But a few remarks:
* This is nontrivial: many generalized-function frameworks (tempered distributions, Colombeau algebras, nonstandard analysis variants) either do not have all these properties or require careful reinterpretation of “classical function”. You must either construct LLL explicitly and prove (C1)–(C3), or show your axioms imply them.
* If you succeed, the result becomes conditional on these new axioms. The Clay Prize requires a proof in standard accepted mathematics; a proof resting on a new foundational axiom must include a convincing, peer-reviewed construction of that axiom system and why it is acceptable (consistent, conservative extension, etc.).

Conclusion for Route 2: Possible but heavy: it displaces the hard part of Clay onto the foundational task of building/justifying the extended algebra.