# What the sixth update adds

The first five posts retain their original wording and evidence. The new result is one supporting proof step: Lean checks that every coefficient is nonnegative **if** the sequence starts at one and satisfies the stated recurrence with nonnegative weights. There is no degree cutoff.

A worker supplied the proof. The coordinator read it, rebuilt it and checked a separate restatement with the index box written explicitly. That restatement checks that the delivered theorem matches the requested statement; it is not a second independent construction of the proof.

The coefficients in this theorem are arbitrary functions under those assumptions. Lean has not yet identified them with the coefficients of the actual rational function used in the analytic argument. Establishing that recurrence and its explicit nonnegative weights is the next obligation.

The [earlier evidence distinctions](review-scope.md) still apply. The [new source](source/OrchardMath/Millennium/HardyRecurrence.lean), [explicit restatement](source/OrchardMath/Millennium/HardyRecurrenceAudit.lean), [axiom output](source/research/millennium/iterations/0030/lean-audit.log) and [handoff summary](lean-recurrence-handoff.json) identify exactly what was checked. The only transitive axioms are `propext`, `Classical.choice` and `Quot.sound`.

This is formalization progress, not a new analytic theorem or a new finite-computation result. It establishes neither originality nor outside human review, and does not solve RH. No production publication is claimed by preparing this follow-up bundle.