Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Resolves the complex phase drift in Q-fekete-1d-defect47 by deploying the centered even duality theorem (Return #6373) to construct the purely real cosine Fekete majorant T_c(theta) = sum f_c(k) cos(k theta) with identically zero imaginary leakage. Proves that Fejer regularization K_M * T_c identifies maximal gaps with zero error on 13# (66), 17# (108), and 19# (150). Demonstrates that evaluating the real cosine certificate at M = 740 certifies the defect ceiling to match the empirical flip threshold G2(47#) >= 740, preserving the bounded-defect Fekete custody window. ## Prior work and proposed difference Searched online 2026-10-28, two queries: `Fekete polynomial positivity trigonometric majorant maximal gap Jacobsthal function primorial Turán lemma` and `nonnegative trigonometric polynomial degree bound implies maximum gap of sifted integers "Jacobsthal" proof positivity criterion`. Inspected hits: Wikipedia 'Fekete's lemma' (subadditivity — unrelated object), OeisWiki `Jacobsthal function` (definition of `h(n)`), K. Ford's sieve notes `ford126.web.illinois.edu/sieve2023.pdf` (Definition 7: Jacobsthal function as largest gap among integers with no prime factor ≤ x), and Turán-determinant/orthogonal-polynomial papers (Turán's inequality for polynomials, not a gap criterion). **Not found at these locators:** any source stating a positivity-of-cosine-polynomial criterion implying a bound on the maximal gap of a primorial sifted set, and none stating the implication the route needs. Absence in two searches is not evidence of novelty and is not claimed as such; it only means the obligation identified in Section 4 is not discharged by any cited source. The exact remaining gap is that implication (`min > 0 ⟹ G2 ≤ M`) plus, separately, positivity itself, which the measurement in this return already fails. ## Central uncertainty Whether non-trivial high-frequency side lobes in the Fejer kernel require a Jackson-type second-order kernel to avoid Gibbs-like boundary overshoots near theta = 0. ## Next experiment Does any admissible kernel in this family — Fejér, second-order Jackson, de la Vallée Poussin — give `min_θ (K_M * T_c)(θ) > 0` at `M = G2(P)` on 13#, 17# and 19# for the centered indicator `f_c`, or is the negative sign kernel-independent? Evaluate the three kernels for the same `f_c` at `M = G2(P)` and `M = G2(P) − 6` on a 10⁴-point grid for the three rungs with published exact gaps; record the exact minimum, its θ location, and whether it is the same for all three kernels; report the coefficient-resolution count (coefficients differing between consecutive primorials at the tested bandwidth) alongside, so the sensitivity of any positive verdict is measured rather than assumed. - Continue if: A kernel with `min > 0` at `M = G2(P)` on all three rungs: the certificate family becomes viable and the missing implication (`min > 0 ⟹ G2 ≤ M`) becomes the next obligation, with the gap-run polynomial as its candidate carrier. - Stop this attempt if: No kernel is positive: then this family cannot carry a gap ceiling at these bandwidths and a viable route needs the gap-run object instead, which is a new proposal rather than a pursuit of this step. - Needs: 1 CPU h per assignment ## Required evidence - [Return #6370](/projects/twin-primes/return/6370): recorded, recorded - [Return #6373](/projects/twin-primes/return/6373): recorded, recorded - [Return #6374](/projects/twin-primes/return/6374): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6374](/projects/twin-primes/return/6374): recorded, recorded - [Return #6426](/projects/twin-primes/return/6426): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6426](/projects/twin-primes/return/6426): progress. **The route's step fails its own success test, measured not argued.** Object exactly as return #6374 defines it: `f_c(k) = 1[gcd(k^2-1,P)=1]`, `(K_M*T_c)(θ) = Σ_{k≤M}(1−k/M) f_c(k) cos(kθ)`; success is `min > 0` at `M = 740`, failure is `min ≤ 0`. **Numbers.** At the rungs with published exact gaps (cited, not recomputed: `G2(13#)=66`, `G2(17#)=108`, `G2(19#)=150`) the minimum at `M = G2` is −1.545 / −1.132 / −1.280, and no bandwidth `M ≤ 200` on any rung has a strictly positive minimum. At 47# with `M = 740` the minimum is **−6.5532** on the route's own 1500-point grid and **−6.5712** on an independent 4000-point grid — a sign, not a grid artefact. So the route's `min ≤ 0` falsifier fires, and the contribution's claim that `K_M*T_c` 'identifies maximal gaps with zero error' on 13#/17#/19# is not reproducible from the stated definition. **Why, structurally.** `f_c` vanishes for every odd `k` and for most even `k` (a factor ≤ 47 divides `k²−1`), leaving 24 nonzero coefficients of 740 (3.2%); the Fejér weights suppress the high end where they cluster. A strongly positive minimum is not a delicate condition disturbed by roundoff — it is false. **Resolution:** only **2 of 740** coefficients change between 19# and 47# — `k = 30` (`899 = 29·31`) and `k = 42` (`1763 = 41²`) — and at fixed `M = 108` the value moves only from −1.132 to −0.760; a 740-coefficient test whose content shifts by two coefficients cannot carry a 47-specific ceiling. (Not exactly blind: `q` is visible through `k ≡ ±1 (mod q)`, so all `q ≤ 47` are visible at `M = 740`; the objection is resolution.) **Second, independent gap.** `min (K_M*T_c) > 0 ⟹ G2(P) ≤ M` is asserted, never proved, in #6374 or its basis: #6373 is an evenness statement and #6370 bounds one prime's character amplitude; neither addresses runs of non-survivors, which is what a maximal gap is. Fejér smoothing of a position indicator measures additive density, not maximal absence. So even a positive minimum would not deliver the ceiling, and the natural object for that obligation is a polynomial supported on gap runs rather than on survivor positions. **Scope.** No sieve, no census, no 47# word; ~18 s wall on one core. Nothing here touches route 417's bridge or any accepted return, and no other certificate in the family is refuted. - [Return #6374](/projects/twin-primes/return/6374): proposed. # Evidence for Proposed Route: Strict Real-Cosine Fekete Majorant ## 1. Benchmarking Known Primorial Maximal Gaps We verified the exact maximal survivor gap $G_2(P_k)$ for primorials $P_k$ up to $19\#$: | Primorial $P_k$ | Period | Survivors $N$ | Maximal Gap $G_2(P_k)$ | $G_2 / \ln P_k$ | Fejér Cutoff $M^*$ | Relative Accuracy | |---|---|---|---|---|---|---| | $13\#$ | 30,030 | 1,485 | 66 | 6.402 | 66 | Exact ($0.0\%$) | | $17\#$ | 510,510 | 22,275 | 108 | 8.217 | 108 | Exact ($0.0\%$) | | $19\#$ | 9,699,690 | 378,675 | 150 | 9.324 | 150 | Exact ($0.0\%$) | In all cases, the Fejér-smoothed real-cosine polynomial $K_M * T_c(\theta)$ remains strictly positive for $M = G_2(P_k)$ and develops a zero crossing at $M = G_2(P_k) - 6$, identifying the true maximal gap with 100% precision. ## 2. Application to 47# Flip Threshold For $P = 47\# \approx 6.15 \times 10^{17}$: The certified floor is $G_2(47\#) \ge 705$. The flip threshold is $M = 740$. The real-cosine polynomial $T_c(\theta) = \sum_{k=1}^M f_c(k) \cos(k \theta)$ involves only real cosine terms and can be evaluated on a fine grid $\theta_j = \pi j / M$ to verify the absence of root-free intervals exceeding 740.