feat(BKLNW): formalize Lemma 10 - #1768
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BKLNW Lemma 10, equations (3.27)-(3.29) of arXiv:2002.11068: if
D_k(a,b) < (k+1)^(k+1) then theta(x) >= x - D_k(a,b) x / (log x)^k on [a,b].
The file imports only Mathlib and no project file, so it can be read and
checked on its own. #print axioms on bklnw_lemma_10, bklnw_lemma_10',
bklnw_lemma_10_paper_form and Dterm_eq_paper each gives
[propext, Classical.choice, Quot.sound]: no sorry, no native_decide.
D_k is formalized pointwise as Dterm k x = (log p)^k (p - theta x)/p with
p = nextPrime x. Dterm_eq_paper proves this equals the paper's printed form
(log p_n)^k (p_n - theta(p_{n-1}))/p_n, and bklnw_lemma_10_paper_form restates
the whole lemma quantified over consecutive prime pairs, so the bridge between
the two shapes is proved rather than assumed.
The (k+1)^(k+1) threshold comes from the tangent-line inequality
(k+1)^(k+1) (t-k) <= t^(k+1), proved by interval_cases over the paper's own
range k = 1..5 with an explicit factorisation for each; this is the only place
k <= 5 is used.
The paper's hypothesis a < b is kept but unused (marked _hab): the argument is
pointwise on [a,b], which is empty when b < a, so the formal statement is very
slightly stronger than the printed one. Keeping it makes the hypotheses match
the paper one-for-one.
No blueprint annotation: @[blueprint ...] needs LeanArchitect, and importing it
would give up the Mathlib-only property. Happy to add it if preferred.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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This PR formalizes Lemma 10 (equations (3.27)–(3.29)) from the BKLNW paper.
#print axiomsonbklnw_lemma_10,bklnw_lemma_10',bklnw_lemma_10_paper_form, andDterm_eq_papereach gives[propext, Classical.choice, Quot.sound].Dterm k x.Dterm_eq_paperproves this equals the paper's printed form, andbklnw_lemma_10_paper_formrestates the lemma over consecutive prime pairs — so the bridge between the two shapes is proved, not assumed.a < bis kept but unused (_hab): the argument is pointwise on[a, b], which is empty whenb < a. The hypotheses match the paper one-for-one.@[blueprint ...]needsLeanArchitectand importing it would give up the Mathlib-only property. Happy to add it if you'd rather have the blueprint entry.This will be used to supply the Case 4 bounds needed for Table 15.
All Lean code and documentation in this PR were produced with substantial AI assistance (Claude, directed by Lingsen Meng), disclosed per PrimeNumberTheoremAnd's PULL_REQUEST_STYLE.md §13.
Co-Authored-By: Claude