Sun Conjecture 5.2: both lowercase strict Turan clauses

Sun, arXiv:2608.13192v1, Conjecture 5.2 asks for both assertions, for every integer n >= 1:

$$ g_n(x)^2>g_{n-1}(x)g_{n+1}(x)\quad\text{for every real }x\leq-1, $$

and

$$ v_n(x)^2>v_{n+1}(x)v_{n-1}(x)\quad\text{for every real }x\leq-1/8. $$

The source's literal initial values and multiplied quadratic/cubic recurrences are reproduced in Library/Recurrence/sun2026generalizations.md. Both endpoints are included. A proof of only one clause, a finite-index check, or a non-strict statement does not settle this source question.

The frozen LucasCompanion.lucasV is an existing repository example of defining a sequence by a recurrence-related algebraic construction. It does not imply either Sun inequality. The new source target tests an all-index strict determinant property for two different literal three-term recurrences over negative real half-lines.

Krasikov's Theorem 8 supplies a weighted non-strict inequality only on the right tail. The g endpoint is degenerate for the local quadratic, and the v clause needs a positive radical normalization and a local interval reaching the exact 1/8 boundary. The uppercase G,V results of Mao--Xiao and the p-adic Li--Sun result do not resolve these clauses. The cited Abreu--Bustoz and Bustoz--Ismail bodies were unread in the bounded source assessment, so no claim about their exact reach is made.

Sequences.g and Sequences.v define the two source sequences. StrictlyIncreasingTail.weighted_turan_nonneg_of_strict_mono proves the reusable strictly-increasing-coefficient tail. GEndpoint gives all-index endpoint strictness by factorial scaling and parity; GStrict.g_strict_turan combines it with local quadratic positivity and the tail for every x<=-1. VTail.v_tail_strict establishes radical coefficient growth, positive scaling, recurrence uniqueness, and strict tail transport. VStrict.v_strict_turan uses 4m^3-(m-1/8)^2(4m+1)=(12m-1)/64>0 to show its local interval reaches the closed 1/8 boundary, then joins it to the tail. In the formal result the order of the two real neighbor factors is reversed from the source's v display by commutativity.

Any positive index and real x in either stated closed domain with a nonpositive corresponding determinant would refute that clause. A failure of positive radical normalization or of the local/tail covering would invalidate this proof route without itself refuting the source conjecture. The theorem's unbounded quantifiers rule out finite search as a substitute.

The primary versioned HTML was checked for both definitions and both Conjecture 5.2 clauses. All six Lean owners have completed joint first Freeze, with twelve canonical accepted/state artifacts. The scoped Lean build exited zero over 3011 jobs, and the symbolic consumer checked the source recurrences, generic strict statements, endpoints, and standard-three axiom closure. Source and companion reviews approved the six-owner result. The prebinding candidate 20a91df388d604fffb8c4436538f94995154283e passed required CI run 36160400233 with all six checks successful.

The GStrict Describe source attaches one typed Proved claim for this slug, with frozen members D5/S1/Recurrence/Sun/GStrict.g_strict_turan and D5/S1/Recurrence/Sun/VStrict.v_strict_turan. The GStrict member is the host; the VStrict member is an additional DeclarationHandle. The five theorem selectors without committed projection fixtures use disclosed, authored Formula statements in their companions. The named-set binding passed local canonical make emit and Scribe projection, Describe, and Markdown checks against the refreshed Lean report. The emitted GStrict page has two v1 member markers under the one claim. Prebinding CI does not certify this later binding change.

theorem; first-tier external two-clause named conjecture. Issue 9833 was filed after the ordinary probes and grants no retrospective open-problem-resolution bind-only exception. Against original joint baseline 15664c84561c0ea57b6ca22cf328802a5ba5f897, the three literal definitions and all five public theorems are assessed together. Each theorem has live content and admission_basis: escape-witness; the two final clauses are separate public results, bound to one problem credit without a conjunction wrapper or duplicate claim. No proof-shape or completion claim is inferred from this problem file itself.

The bounded prior-art search did not find a later exact proof through 25 September 2026; worldwide absence and priority are unverified. The source mathematical result and repository delivery completion are different questions. Final candidate CI, ordinary merge to dev, main synchronization, and a separate completion audit remain to be checked by the delivery owner.