The maximal variance of a unilaterally truncated chi distribution: the negative-dimension regime, two sharp Mills-ratio inequalities, and a certified table
Abstract
For a radially symmetric Gaussian in n dimensions restricted to the shell R ≥ a, Petrella (arXiv:2511.11566) writes the variance of R at a fixed mean M as M² V(n,r) with r = a/σ, x = r²/2 and V(n,r) = Γ(frac n2,x)Γ(frac(n+2)2,x)/Γ(frac(n+1)2,x)² - 1, an expression that makes sense for every real n once r > 0, and conjectures on numerical evidence that V(n, ·) is maximal in the limit r → 0 for every n in -100 ≤ n ≤ 512. We prove the conjectured maximum in the negative-dimension regime: for every integer n ≤ -3 and every a > 0, V(n,a) < 1/(n(n+2)), the value the source derives for the limit; the bound is never attained, and the constant is sharp (certified at n = -3 and n = -6, and for every integer n ≤ -5 through an explicit lower bound tending to 1/(n(n+2))). The mechanism is that Wₖ(a) = (k-1)aᵏ⁻¹∫ₐ^∞ x⁻ᵏe^(-x²/2) dx is a strictly concave, hence log-concave, function of the integer order k; in incomplete-gamma terms this is the sharp reverse Turán inequality Γ(ν-1/2,x)Γ(ν+1/2,x)/Γ(ν,x)² ≤ ν²/(ν²-1/4) at half-integer ν ≤ -1, whose unit-shift analogue is a theorem of Baricz and Ismail. For n ≥ 1/2 the conjecture is a corollary of a 2023 lemma of Cao and Vempala on truncations of log-concave densities; we record the reduction, and at n = 1 and n = 2 we give machine-checked proofs over the whole half line in the form of the sharp Mills-ratio inequalities Q(a)² + aQ(a) ≤ π/2 and (a + Q(a))² ≥ (π/4)(a²+2). Twelve cells of the source's table are certified as rational enclosures, its r = 0 column is given in closed form for n = 1,…,6, and one printed value is corrected: at n = 6, r = 0 the spread is 16/(15√(2π)) = 0.42554, not 0.42553. The strip 0 < n < 1/2 remains open. Every theorem is verified in Lean 4 with Mathlib, with no axioms beyond the three standard ones.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-09-07 03:53 UTC
File fingerprint
8605d308b8ddfa62fd5be78ee341311727359d0b286c57185bddbb8cf788793b
Claim ledger
Stated results
TCV1known2026-09-03
a*Q(a) <= 1 for a >= 0, from the nonnegativity of intₐⁱnf (x-a) e^(-x²/2) dx
TCV2known2026-09-03
Q(a)² + a*Q(a) >= 1: the one-degree-of-freedom truncated variance is nonnegative
TCV3known2026-09-03
closed form V(n=1, a) = Q²+aQ-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV4known2026-09-03
closed form V(n=2, a) = (a²+2)/(a+Q)²-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV5known2026-09-03
closed form V(n=3, a) = (a³+3a+3Q)(a+Q)/(a²+2)²-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV6known2026-09-03
closed form V(n=4, a) = (a⁴+4a²+8)(a²+2)/(a³+3a+3Q)²-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV7known2026-09-03
closed form V(n=5, a) = (a⁵+5a³+15a+15Q)(a³+3a+3Q)/(a⁴+4a²+8)²-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV8known2026-09-03
closed form V(n=6, a) = (a⁶+6a⁴+24a²+48)(a⁴+4a²+8)/(a⁵+5a³+15a+15Q)²-1 from the defining moment integrals intₐⁱnf xᵐ e^(-x²/2) dx
TCV9known2026-09-03
Q(0) = sqrt(2*pi)/2
TCV10known2026-09-03
the source's explicitly unproven maximality claim at n = 1, for ALL a >= 0: Q(a)² + a*Q(a) <= pi/2, with equality at a = 0
TCV21known data2026-09-03
source table cell n = 1, |r| = 1/2: 0.2061968 < V < 0.2061969 (the source prints 0.20620)
TCV22known data2026-09-03
source table cell n = 2, |r| = 1/2: 0.1877242 < V < 0.1877243 (the source prints 0.18772)
TCV23known data2026-09-03
source table cell n = 3, |r| = 1/2: 0.1565793 < V < 0.1565794 (the source prints 0.15658)
TCV24known data2026-09-03
source table cell n = 6, |r| = 1/2: 0.0862826 < V < 0.0862827 (the source prints 0.08628)
TCV25known data2026-09-03
source table cell n = 1, |r| = 1: 0.0855952 < V < 0.0855953 (the source prints 0.08560)
TCV26known data2026-09-03
source table cell n = 2, |r| = 1: 0.0943813 < V < 0.0943814 (the source prints 0.09438)
TCV27known data2026-09-03
source table cell n = 3, |r| = 1: 0.0977226 < V < 0.0977227 (the source prints 0.09772)
TCV28known data2026-09-03
source table cell n = 6, |r| = 1: 0.0801326 < V < 0.0801327 (the source prints 0.08013)
TCV29known data2026-09-03
source table cell n = 1, |r| = 2: 0.0202904 < V < 0.0202905 (the source prints 0.02029)
TCV30known data2026-09-03
source table cell n = 2, |r| = 2: 0.0233618 < V < 0.0233619 (the source prints 0.02336)
TCV31known data2026-09-03
source table cell n = 3, |r| = 2: 0.0266677 < V < 0.0266678 (the source prints 0.02667)
TCV32known data2026-09-03
source table cell n = 6, |r| = 2: 0.0363631 < V < 0.0363632 (the source prints 0.03636)
TCV33known data2026-09-03
exact untruncated value V(n = 1, r = 0) = (pi-2)/2
TCV34known data2026-09-03
exact untruncated value V(n = 2, r = 0) = 4/pi - 1
TCV35known data2026-09-03
exact untruncated value V(n = 3, r = 0) = 3*pi/8 - 1
TCV36known data2026-09-03
exact untruncated value V(n = 4, r = 0) = 32/(9*pi) - 1
TCV37known data2026-09-03
exact untruncated value V(n = 5, r = 0) = 45*pi/128 - 1
TCV38known data2026-09-03
exact untruncated value V(n = 6, r = 0) = 768/(225*pi) - 1
TCV39known data2026-09-03
exact spread sigma(n = 6, r = 0) = 16/(15*sqrt(2*pi))
TCV40correction2026-09-03
the source's printed sigma = 0.42553 at n = 6, r = 0 is a truncation, not a rounding: 0.4255384 < sigma < 0.4255385, so the correct five-decimal value is 0.42554
TCV41known2026-09-03
control: a claimed maximum pi/2 - 1 - 1/1000 fails, because V(1,0) = (pi-2)/2 is attained
TCV42known2026-09-03
control: V(1, r=1) < V(1, r=0): the maximality theorem is not the statement that V(1,.) is constant
TCV43known2026-09-03
control: the source's n = 3, |r| = 1 cell is not 0.09773 (a too-large claim fails)
TCV44known2026-09-03
control: V(1,0) > 0: at a = 0 there is no truncation and the value is the untruncated chi CV², not zero
TCV11known2026-09-03
the source's claim at n = 2 as a Mills-ratio inequality: (a + Q(a))² >= (pi/4)(a²+2) for all a >= 0, with equality at a = 0
TCV12known2026-09-03
the source's claim at n = 2 in its own terms: V(2,a) <= 4/pi - 1 for all a >= 0, the value Vₘax,r(2) it prints
TCV50prose2026-09-03
PROSE: for every real n and every x > 0 the ratio Gamma(n/2,x)Gamma((n+2)/2,x)/Gamma((n+1)/2,x)² is non-increasing in x; equivalently the coefficient of variation of the left-truncated chi distribution is maximal at truncation 0. This is the source's claim on its whole stated range -100 <= n <= 512, including the negative-dimension regime its numerics only sampled
This ledger entry is reported in prose and is not bound to a Lean theorem.TCV60candidate2026-09-03
for every integer n <= -3 and every truncation a > 0, V(n,a) <= 1/(n(n+2)): the source's conjectured maximal variance, at the source's own limiting value, in the negative-dimension regime
TCV61routine2026-09-03
the r = 0 cell 0.33333 of the source's Table 4 (p. 19) at n = -3 is the supremum: V(-3,a) <= 1/3 for every a > 0
TCV62routine2026-09-03
the r = 0 cell 0.04167 of the source's Table 4 (p. 19) at n = -6 is the supremum: V(-6,a) <= 1/24 for every a > 0
TCV63candidate2026-09-03
the mechanism: Wₖ(a) = (k-1) aᵏ⁻¹ intₐⁱnf x⁻ᵏ e^(-x²/2) dx is concave in the order k (its second difference is intₐⁱnf aᵏ⁻² x^(-(k-1)) (x-a)² e^(-x²/2) dx >= 0); equivalently nu -> -nu x⁻ⁿᵘ Gamma(nu,x) is positive and concave on nu < 0 for every x > 0, whence the sharp reverse-Turan inequality Gamma(nu-h,x)Gamma(nu+h,x)/Gamma(nu,x)² <= nu²/(nu²-h²) at EVERY step h, for nu < -h < 0
TCV64routine2026-09-03
the concavity of W in the order is strict, the defect being intₐⁱnf aᵏ⁻² x^(-(k-1)) (x-a)² e^(-x²/2) dx > 0
TCV65routine2026-09-03
the maximum is never attained: V(n,a) < 1/(n(n+2)) strictly, for every integer n <= -3 and every a > 0
TCV66known2026-09-03
0 <= V(n,a) for every integer n <= -3 and a > 0: Cauchy-Schwarz for the negative-order tail moments, so TCV60 pins a nonempty interval
TCV67routine2026-09-03
sharpness at n = -3: 1/3 - 1/500 < V(-3, 1/1000), so the constant 1/3 cannot be lowered
TCV68routine2026-09-03
sharpness at n = -6: 1/24 - 1/100000 < V(-6, 1/1000), so the constant 1/24 cannot be lowered
TCV71routine2026-09-03
quantitative sharpness for every integer n = -(i+5) <= -5 and 0 < a <= 1: V(n,a) >= (i+4)²/((i+3)(i+5)) (1 - a²/(i+1))(1 - a²/(i+3)) - 1, whose a -> 0 limit is 1/(n(n+2))
TCV69routine2026-09-03
control (a too-small constant fails): 'for all a > 0, V(-3,a) <= 1/3 - 1/500' is refuted
TCV70routine2026-09-03
control (a too-large claim fails): 1/3 <= V(-3,a) is false at every a > 0
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. R. J. Petrella, *The Maximal Variance of Unilaterally Truncated Gaussian and Chi Distributions*, arXiv:2511.11566 v1 (14 Nov 2025, math.ST, 51 pp). v1 only, not withdrawn, no journal-ref; OpenAlex W4416331503, 0 citations.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7