The sharp one-sided tail bound for integer-valued random variables under a fourth-moment constraint

Lattice weak duality: a dual polynomial for the tail functional need only be nonnegative AT THE INTEGERS, and both certificates used here are negative somewhere on R

The sharp one-sided Chebyshev constant for integer-valued X: sup P(X >= t) = 1/(t(t+1)) for every integer t >= 1 with E X = 0 and E X² = 1, against Cantelli's 1/(1+t²)

The binding regime of the lattice four-moment map: V1^Z(t,kappa) = (kappa-1)/(t²(t²-1)) for every integer t >= 2 and every 1 <= kappa <= t²-t+1, extremal law on -1,0,1,t, certificate x²(x²-1)/(t²(t²-1))

The saturated regime of the lattice four-moment map: V1^Z(t,kappa) = 1/(t(t+1)) for every integer t >= 1 and every kappa >= t²-t+1, the threshold being exactly the fourth moment of the two-moment extremal law

The lattice constant is strictly below Cantelli's for every integer t >= 2, and equal to it at t = 1

Negative controls: a too-large and a too-small claim refuted at (t,kappa) = (2,2), and each of the three regime hypotheses (t >= 2, kappa <= t²-t+1, kappa >= 1) shown not to be decoration

The integrality gap against the parent family, exhibited on both sides: the lattice class is a subclass of arXiv:2607.05226's class, Cantelli's extremal law at t=2 is an explicit member of the continuous class attaining 1/5, and 1/5 is provably unattainable on the lattice, where the maximum is 1/6

The a-priori support bound: a law of the class puts mass at most kappa/N⁴ beyond radius N

The map at every real threshold: an integer-valued law has P(X >= t) = P(X >= ceil t), so V1^Z(.,kappa) is a STEP function in t, constant on each (n-1,n], and the two closed forms of M3/M4 determine V1^Z on all of (0,infinity) x [1,infinity)

The lattice tail staircase at arbitrary variance: sup P(X >= s) = (v + a(a+1))/((s+a)(s+a+1)) with a = floor(v/s), for integer-valued X with E X = 0 and E X² = v; piecewise linear in v with breakpoints exactly at the multiples of s; stated on an arbitrary grid hZ

The exact comparison with Cantelli: the identity (v + a(a+1))(v+s²) - v(s+a)(s+a+1) = (v-as)(v-(a+1)s), hence the lattice constant is at most Cantelli's v/(v+s²) with equality if and only if s divides v, plus the uniform envelope v/(v+s²) - V^Z(s,v) <= 1/(4(v+s²))

Grid refinement: the sharp constant on (1/k)Z at every real threshold; monotone along divisibility chains but NOT in the mesh (V on half-Z at t=2 is 1/5, strictly above V on third-Z which is 11/56); equal to Cantelli's 1/(1+T²) exactly when T divides k; and a gap at most 1/(4k²(1+T²)) at integer thresholds T

Negative controls for the staircase: too-large and too-small at (s,v) = (2,3), the step index shown not to be decoration on both sides (a = 0 gives 1/2 and a = 2 gives 9/20, both refuted against the true 5/12), and Cantelli's 1/5 shown unattainable on third-Z but attainable on half-Z

The lattice six-moment map: for every integer t >= 2 and all kappa, lambda >= 1, sup P(X >= t) = min 1/(t(t+1)), (kappa-1)/(t²(t²-1)), (lambda-1)/(t²(t⁴-1)), attained; three regimes with the arithmetic thresholds kappa* = t²-t+1 and lambda* = t⁴-t³+t²-t+1 (the fourth and sixth moments of the two-moment extremal law -1,0,t) and the straight-line boundary lambda = 1 + (t²+1)(kappa-1); certificate x²(x⁴-1)/(t²(t⁴-1)), extremal support unchanged at -1,0,1,t

The even-moment hierarchy collapses on the lattice: for ANY finite set J of even-moment bounds E X^(2(i+1)) <= cᵢ imposed on an integer-valued X with mean 0 and variance 1, sup P(X >= t) = min(1/(t(t+1)), min_(i in J) (cᵢ-1)/(t^(2(i+1)) - t²)), attained by the SAME four-atom law on -1,0,1,t for every J – the extremal support is frozen and the map is a minimum of affine functions

The sharp discrete (m2,m4,m6) cone: E X⁶ >= (a²+(a+1)²) E X⁴ - a²(a+1)² E X² for integer-valued X and every integer a >= 0, sharp at a = floor(sqrt kappa) with the three-atom law on 0,a,a+1; the excess over Cauchy-Schwarz is the exact product lambdaₘin(kappa) - kappa² = -(kappa-a²)(kappa-(a+1)²), and equality holds iff kappa is a perfect square

The step ratio in closed form: on each step (n-1,n] the lattice map is constant and the continuous map of arXiv:2607.05226 restricted to t >= 1 is min(1/(1+t²), (kappa-1)/((t²-1)²+kappa-1)), continuous and strictly decreasing; the supremum of their ratio over the step is the three-branch closed form Rₙ(kappa) at the excluded left endpoint and is NOT attained; the minimum rhoₙ at t = n is attained; Rₙ(kappa) < (n²-1)/(n-2)² for every kappa > 1 and n >= 3, while R₂(kappa) = 6/(kappa-1) on 1 < kappa <= 3 is unbounded

Negative controls for the six-moment map: the README's guessed certificate x²(x²-1)(x²-4) is lattice-nonnegative but NOT dual feasible, and the bound it would give is refuted at (t,kappa,lambda)=(3,20,61); its sign repair x⁶-6x²+5 is strictly weaker for every t >= 2; each of the three regime windows is refuted outside itself; t >= 2, lambda >= 1 and variance 1 are each shown not to be decoration