Local stability of Shapiro–Diananda cyclic sums: explicit failures and exact diagonal identities

Diananda inequality S_(n,k) ≥ n/k fails at the smallest saddle cell of each 3 ≤ k ≤ 10, with explicit positive-integer witnesses; (7,4) is the smallest cell of the table

Further explicit failures: (14,3), (17,3), (20,3), (11,4) and the source's n ≥ 25 threshold at (25,3), (25,4), (25,5)

n = k+1: Q_(k+1,k)(u) = ((k+1)/k³)‖u‖² on ∑u = 0, exactly, for every k ≥ 1; hence Cˡoc_(k+1,k) = (k+1)/k³

n = k+2: Q_(k+2,k)(u) = ((k+2)/(2k³)) ∑ᵢ(uᵢ+uᵢ₋₁)², exactly, for every k ≥ 1; so Cˡoc ≥ 0 always, = 0 iff k even

n = 2k+1: Q_(2k+1,k)(u) = (1/(2k²))‖u‖² + (1/k³)∑ᵢLᵢ², hence Cˡoc_(2k+1,k) ≥ 1/(2k²) > 0 — the equal point of S_(2k+1,k) is a strict local minimum for every k ≥ 1

n = 2k+2: Q_(2k+2,k)(u) = (1/(4k²))∑ᵢ(uᵢ+uᵢ₊ₖ₊₁)² + (1/k³)∑ᵢLᵢ², so Cˡoc ≥ 0 for every k

n = k+3: k³Q(u) = (k+3)‖u‖² + (k+4)∑uᵢuᵢ₋₁ + (k+2)∑uᵢuᵢ₋₂, whose symbol factors as 2(k+2)(c+1/(k+2))(c+½) in c = cos θ

Controls: the equal point gives exactly n/k at (7,4), (11,3), (13,4), and the source's periodic equality family gives exactly 3 at n = 12, k = 4, d = 4

Negative controls: the gap at (7,4) is exactly 3/1100 and a gap > 1/300 is refuted; the gap at (11,3) is exactly 15/5168; two one-step perturbations of the (7,4) witness fail; at (13,4) and (9,4) the same construction gives a value above n/k

Q_(n,k)(1,…,1) = 0 for every n, k ≥ 1 (the scaling direction is null), and hence the n = k+1 identity fails without its ∑ u = 0 hypothesis

Explicit zero-sum rational saddle directions certifying Cˡoc < 0 at (11,3), (7,4), (9,5): Q = -1/3, -3/32, -3/25

k = 11, n = k+3 = 14: Q_(14,11)(u) = -10/1331 < 0 — past the source's k ≤ 10 table, on the diagonal that carries the first failure for every 11 ≤ k ≤ 24

Erratum to arXiv:2606.05504v1: the value -0.005046 printed for Cˡoc_(17,3) is below the infimum over [-1,1] of the source's own symbol Γ₃(c) = 2(1-c)(2c+1)(3c+2)/27, whose minimum is -0.00489021… at the root (-1-√91)/18 of 18c²+2c-5; the correct value is -0.004681049…. The n = 19 and n = 20 entries (0.002959 and -0.004826) are also wrong (0.001402823…, -0.004886585…)

The exact classification for 3 ≤ k ≤ 24 and all n (finite check to n = 12k plus the source's n > 12k theorem): for every 11 ≤ k ≤ 24, Cˡoc_(n,k) > 0 exactly at n ∈ k+1, 2k+1 for even k and n ∈ k+1, k+2, 2k+1, 2k+2 for odd k, and the first failure is at n = k+3

Every cell of the Boarder–Daykin 1973 list of cases undecided for k ≤ 12 is a strict local minimum or quadratically degenerate; no saddle cell appears on it, and every saddle cell with k ≤ 12 is absent from it

γ_(m,k) = 0 ⟺ λ_(m,k) = 0 ⟺ (n/gcd(m,n)) ∣ k, verified exactly for every k ≤ 40 and, at each such k, for every order d ≥ 2 whatsoever: quadratic degeneracy of the equal point has only the periodic-equality explanation

Universal lower bound from completing the square: Q_(n,k)(u) + (1/(4k))‖u‖² = (1/k³) ∑ᵢ (Lᵢ - (k/2)uᵢ)² for every n, every k ≥ 1 and every u (no zero-mean hypothesis); hence Cˡoc_(n,k) ≥ -1/(4k) for every (n,k)