Source Clarifications: Accuracy-Diversity

Clarified regularity conditions

The following source-consistent, natural model conventions state the shared regularity domains. The report therefore describes only the affected results as exact with these conditions; this presentation does not erase the result-local source corrections below.

For Corollary 1 and Theorem 1(ii)–(iv), every named preferred type has positive probability: $p_t>0$ . These are the all-coordinate share formulas; a zero-probability type calls for a separate supportwise formulation.
In the fixed-top- $k$ model of Theorem 1(i)–(ii), conditional values are nonnegative almost surely. This identifies the finite allocation objective with the source’s exact top- $k$ value primitive. Theorem 1(i)’s separate nondegenerate-top-value repair is not absorbed by this condition.
For Proposition 4, write the preference measure as normalized sphere volume with an a.e.-measurable density that is positive almost everywhere, and take the nonconstant radial kernel to be continuous with values in $(0,1]$ . The density need not be continuous: it transfers sphere-volume null sets to the preference law, while kernel continuity rules out a pointwise Laplace supremum supported only on a null spike.

Value and type-support restrictions

Theorem 1(i)’s finite-discrete branch additionally needs a positive top value, a strictly smaller nonnegative second bound, and positive law mass at the top and below it. A point-mass law makes the fixed-top- $k$ objective eventually flat and does not force uniformity of every optimal sequence.

Proposition 2 and the finite uniform model

Relaxed allocation $a_t=N s_t-1$ → $(N+T)s_t-1$ , where $s_t=\sqrt{p_t}/\sum_j\sqrt{p_j}$ , $p_t$ is type weight, $T$ the type count, and $N$ the budget. The printed coordinates sum to $N-T$ , not $N$ .
Printed rounding error $(T+1)/N$ → checked bound $(2T+1)/N$ on $|a_t/N-s_t|$ in the positive top- $k$ domain, where $k$ is the consumption count. A compiled strictly-positive-PMF witness refutes the sharper printed bound. Both bounds give the same square-root share limit.

Theorem 2: independent rank-varying Bernoulli values

“Identically distributed” rank coordinates → independent Bernoulli coordinates with rank-dependent probability $c((i+1)+d)^{-\alpha}$ , where $i$ is the zero-based rank, $c$ the scale, $d$ the rank shift, and $\alpha$ the decay exponent. Top-one results use $c>0$ , $d\ge0$ , and first probability strictly below one: flat cases $c=0$ or first probability one need not force universal homogeneous shares.
The all-consumed $p_t^{1/\alpha}$ formula at $\alpha=0$ → choose a type of maximal $p_t$ separately at zero exponent. The positive-exponent power law retains its stated domain; cross-type independence is unnecessary because the objective conditions on the selected type.

Here $h(a)$ is one type’s expected top- $k$ value from an allocation of $a$ items (or the abstract one-type objective in D.1), and $p_t$ is type $t$ ’s selection probability.

The all-type finite-discrete Theorem 1(i) route explicitly uses positive type weights and a nonnegative common value law with a positive top value, a strictly lower nonnegative second bound, and positive mass at the top and below it. These conditions support divergence of every optimal allocation coordinate; the present theorem does not establish an all-coordinate limit without them.
D.1(i)’s $B>0,\ \sigma<0$ in $\log(A-h(a))/(B a^\sigma)\to1$ → $B<0,\ \sigma>0$ and eventually $A-h(a)>0$ , describing approach to saturation $A$ . Under the printed signs, weights $(1/4,3/4)$ , $h(a)=1-1/(a+1)$ , $A=2$ , $B=1$ , and $\sigma=-1$ satisfy the limit but give square-root shares rather than uniform shares.
D.1’s power-branch comparison of original objectives → comparison of deficits $A-h(a)$ ; a strict deficit ratio need not reverse the objective ratio as both objectives tend to $A\sum_t p_t$ . The logarithmic branch maximizes, rather than minimizes, $\sum_t p_t\log(x_t)$ , where $x_t$ are allocation shares. The logarithmic and sublinear-power branches use strict discrete concavity, and all four branches require positive weights for asserted coordinates.

Order statistics and integer rounding

Exponential/Pareto marginal and strict-concavity statements → eventual statements on valid order-statistic ranks. The exponential rate is $\lambda>0$ ; for fixed upper-rank offset $r$ , $\mu(q-r,q)-\log(q)/\lambda\to(\gamma_{\mathrm{EM}}-H_r)/\lambda$ , where $\mu$ is the expected order statistic and $H_r$ the harmonic number; $\gamma_{\mathrm{EM}}$ is the Euler–Mascheroni constant. The Pareto density is $\alpha x^{-\alpha-1}$ on its support, with $\alpha>1$ .
Lemma D.5’s convex maximization wording → strictly concave maximization. For real/integer fixed-sum maximizers $x,a$ across $m$ coordinates, the checked window is $\lfloor x_t\rfloor<a_t+m$ and $a_t<\lfloor x_t\rfloor+m$ . A derivative-free strict-concavity comparison supplies the rounding argument.

Proposition 4

Equation (18)’s membership in an infimum → the real inequality $\Gamma(\pi)\le\Gamma(\alpha)$ for every profile $\alpha$ , with $\pi$ uniform and $\Gamma$ the source objective. Equation (20)’s unweighted surface integral → the preference-weighted measure in Equation (17); full support does not equate those finite integrals.
Under the clarified regularity conditions above, the Proposition 4 endpoint is exact. The density condition and radial continuity have distinct roles; neither asserts continuity of the preference density. The separate Equation
1. type correction and Equation (20) finite-measure repair remain visible in the preceding bullet.

All-consumed and Bernoulli endpoints

Theorem 1(v)’s maximal-type-weight optimizer → a weak optimizer when the common conditional mean is nonnegative; uniqueness requires positive mean. An all-consumed item contributes its type weight times that mean.
Theorem 3’s log-share formula → $0<q_t<1$ for every Bernoulli parameter, so $1/\log(1/(1-q_t))$ is finite and defined.
Corollary 3’s $q>0$ → $0<q<1$ for its universal share conclusion. With at least two positively likely types and $q=1$ , giving each one item attains success probability one and permits nonuniform optimal sequences.