The third-moment Rademacher stability constant is exact on the flat family, on the spike region and on the fourth-moment region Σᵢaᵢ⁴ ≥ 1/2

the source's dimension-three witness is an exact equality case: at a = (1/√2, 1/√2, 0) in ℝ³, E|S̄₃|³ − E|Σ aᵢεᵢ|³ = (5√3 − 6√2)·Δ₃(a) with Δ₃(a) = 1/6, so C₃ᵒpt ≤ 5√3 − 6√2

the extremal dimension step is exact: E|S̄₃|³ − E|S̄₂|³ = (5√3 − 6√2)(1/2 − 1/3), i.e. the consecutive chord slope r₂ equals 5√3 − 6√2 with no slack

the chord bound at every pair of dimensions: for all 1 ≤ k < n, E|S̄ₙ|³ − E|S̄ₖ|³ ≥ (5√3 − 6√2)(1/k − 1/n) — the source's conjectured inequality for the flat-on-k coefficient vectors, proved for every n

sharpness of the chord bound: it is STRICT for every 1 ≤ k < n except (k,n) = (2,3), where it is the equality of RC2 — so 5√3 − 6√2 is the unique extremal chord slope of the third-moment sequence

the source's conjecture holds on the spike region at every dimension: if some coordinate satisfies Σ_(j≠i₀)|aⱼ| ≤ |a_(i₀)| then E|Σ aᵢεᵢ|³ ≤ E|S̄ₙ|³ − (5√3 − 6√2)Δₙ(a) for every n ≥ 2 — in particular for every coefficient vector with at most two nonzero entries

negative controls: the constant 18/100 already fails at the dimension-three witness; the spike hypothesis is non-vacuous (the witness satisfies it with Δ₃ = 1/6 > 0); the spike moment identity fails without the dominance hypothesis (flat three-vector: 5√3/6 ≠ 7√3/9); and the chord bound is strict at (k,n) = (4,5)

the source's infimum restricted to the spike region is EXACTLY 5√3 − 6√2: the lower bound holds at every n ≥ 2 and the dimension-three witness attains it

the sharp spike constant in dimension four is exactly 6 − 4√2 = 0.343145750…: the inequality holds for every spike vector of ℝ⁴ and the flat-on-two vector attains it

the closed form of the third absolute moment: 2ⁿE|Sₙ|³ = 8k²·C(2k,k) for n = 2k and (k+1)(4k+1)·C(2k+2,k+1) for n = 2k+1, equivalently E|S̄₂ₖ|³ = 2√(2k)·βₖ and E|S̄₂ₖ₊₁|³ = (4k+1)βₖ/√(2k+1) with βₖ = C(2k,k)/4ᵏ; plus 16ᵏ ≤ 4k·C(2k,k)²

measured: the exact algebraic candidates for the fixed-dimension optimum Φₙ = inf over the unit sphere of ℝⁿ — Φ₄ = 6−4√2, Φ₅ = 27√5/2−30, Φ₆ = 15√6/8−3√2, Φ₇ = 195√7/8−105√6/4, Φ₈ = √2/4, Φ₉ = 1785/16−315√2/4, Φ₁₀ = 315√10/256−5√2/2, Φ₁₁ = 6615√11/128−3465√10/64, Φ₁₂ = 693√12/640−12√2/5 — from the flat-family minimum, matched by direct sphere minimisation for n ≤ 8

zero-padding invariance of E|Σ aᵢεᵢ|³, and with it the source's conjectured inequality at EVERY flat-on-k vector of ℝⁿ (coordinates ±1/√k on an arbitrary k-element set, 0 off it) for all 1 ≤ k ≤ n — the literal reading of the chord bound RC3, together with the permutation and sign invariances it needs

the conjecture of arXiv:2608.17802v1 holds with its SHARP constant on the ℓ⁴ region Σᵢ aᵢ⁴ ≥ 1/2 of the unit sphere, at every n ≥ 2, and the restricted infimum there is EXACTLY 5√3 − 6√2 — and exactly 6 − 4√2 in dimension four — new content: n ≥ 4 and uniformity in n; the n = 3 case is the source's Theorem 1.4

measured, for the parked full-sphere problem in dimension four: the three sign cells of ℝ⁴ and their exact cubics (T₃ = 3a₁−2a₁³ on u ≥ 0, minus (1/4)u³ on u ≤ 0 ≤ v, minus (1/4)(u³+v³) on v ≤ 0, sorted, u = a₁−a₂−a₃−a₄, v = a₁−a₂−a₃+a₄); the deficit identity D = (a₁−1/√2)²·Q_C(a₁) + 2C·(a₂²a₃²+a₂²a₄²+a₃²a₄²) + (1/4)u³₋ + (1/4)v³₋; that D ≥ 0 has TWO zeros on the sorted sphere, (1/√2,1/√2,0,0) and the flat vector, so every certificate on q < 1/2 is tight at both; that min Σaᵢ⁴ over the spike region is 1/2 at n = 3..7; and the counted SOS price of the remainder — a 35 × 35 Gram per cell with √2 carried as a variable (s² = 2, face over ℚ(√2)), or 70 × 70 after the rational substitution 12AB − A² − 4B² ≥ 0, in both cases on a proper face that the alternating-projection solver cannot reach