Answer: \(E^*(\varepsilon)=1+2\log(1/\varepsilon)\).
Theorem 3.4 (Upper Bound for Multiplicative Envy) — class: algorithmic
Given a finite set of complete valuation types \(\Theta\) with masses \(\mu\), an online item stream with values \(v_\theta(s)\in\{0\}\cup[\varepsilon,1]\), and per-capita supply scaled with population size, consider the fluid limit of the paper's greedy rule. Track a measure over individual bundle states, including each type's value for each agent's bundle, and route arriving item flow to states maximizing the paper's priority \(v_\theta(s)/u\). When every type has unbounded monopolistic utility, what is the supremal asymptotic multiplicative envy \(\operatorname*{ess\,sup}_{x,y} w_{\theta(x)}(y)/w_{\theta(x)}(x)\) over positive-mass states and admissible streams? Is it \(1+2\log(1/\varepsilon)\)?
Fix \(\varepsilon\in(0,1]\). A society is specified by its nonempty support \(\Theta\): the list of its \(\tau\) complete valuation types and their positive binary-rational masses \(\mu_\theta\), with \(\sum_\theta\mu_\theta=1\). Here \(\tau\) is the support size, not the number of possible types. All individuals have equal positive weight.
There are \(m\) item kinds, with binary-rational additive values \(v_\theta(j)\in\{0\}\cup[\varepsilon,1]\); each kind is valued positively by some supported type. An adversarial stream \(j(t)\) is piecewise constant with finitely many phases on each bounded interval and supplies one item per unit population per unit time. Require \(\int_0^T v_\theta(j(t))\,\mathrm dt\to\infty\) for every \(\theta\).
The population state is a probability measure \(\nu_t\) on \(\Theta\times\mathbb R_+^\tau\). In state \((\theta,z)\), coordinate \(z_\eta\) is type \(\eta\)'s value for that individual's bundle, and their own utility is \(z_\theta\). Initially \(\nu_0=\sum_\theta\mu_\theta\delta_{(\theta,0)}\).
A greedy trajectory has a nonnegative routing measure \(r_t\ll\nu_t\) of total mass one, concentrated on states attaining the \(\nu_t\)-essential maximum of \(v_\theta(j(t))/z_\theta\). Ties may be resolved arbitrarily using only current and past information; interpret a positive number divided by zero as \(+\infty\), and \(0/0=0\).
Receiving an item remains an individual bundle jump: with \(\mathbf v(j)=(v_\eta(j))_{\eta\in\Theta}\), require \(\frac{\mathrm d}{\mathrm dt}\int f\,\mathrm d\nu_t=\int[f(\theta,z+\mathbf v(j(t)))-f(\theta,z)]\,\mathrm dr_t\) for bounded test functions, with \(\nu_t\) remaining nonnegative. Thus the measure evolves continuously while individual items remain indivisible.
Define \(E(t)=\operatorname*{ess\,sup}_{((\theta,z),(\phi,z'))\sim\nu_t\otimes\nu_t}z'_\theta/z_\theta\), using the same zero-denominator conventions, and \(E^*(\varepsilon)=\sup\limsup_{t\to\infty}E(t)\), where the supremum ranges over all finite supported societies, finite item menus, admissible streams, and greedy trajectories. Any algorithmic running times should be stated in \(m,\tau,L\), the number of stream phases supplied, and other relevant parameters, where \(L\) is the binary input length.
What is the sharp value of \(E^*(\varepsilon)\)? Is \(E^*(\varepsilon)=1+2\log(1/\varepsilon)\), or does the continuous society have a different worst-case asymptotic multiplicative envy factor?
For finite agents, integral greedy has asymptotic multiplicative envy at most \(1+2\log(1/\varepsilon)\), with matching lower-bound instances. [Yang, Liao, and Kroer](https://arxiv.org/pdf/2308.09277)
A later full version gives the upper-bound error as \(O(n/V_i)\), where \(V_i\) is the envious agent's monopolistic utility, and retains the matching lower bound; no result settling the bundle-state measure dynamics above was located. [Yang, Liao, Gao, and Kroer](https://arxiv.org/pdf/2408.02403)
The bundle-state measure dynamics have the same sharp asymptotic factor:
\( \displaystyle E^*(\varepsilon)=1+2\log(1/\varepsilon). \)
The proof below applies directly to the measure dynamics. In particular, it does not require a limit theorem connecting finite populations to continuous societies.
Write \(C_\varepsilon=1+2\log(1/\varepsilon)\), with the natural logarithm. The finite-agent result and its lower-bound construction appear in [Yang, Liao, Gao, and Kroer, Section 7 and Appendix D](https://arxiv.org/pdf/2408.02403). Here we establish a population-independent upper bound and construct the required lower bound with a fixed finite menu and a single infinite stream.
We use the forward equation in its usual time-integrated sense. All indicator functions below are bounded measurable test functions.
For each supported type \(\theta\), define its monopolistic utility and its essential minimum individual utility by
\( \displaystyle V_\theta(t)=\int_0^t v_\theta(j(s))\,\mathrm ds,\qquad a_\theta(t)=\operatorname*{ess\,inf}_{(\theta,z)\sim\nu_t}z_\theta. \)
The essential infimum is taken within the type-\(\theta\) population.
Type masses remain constant. Applying the evolution equation to the indicator of a type gives
\( \displaystyle \nu_t(\{\theta\}\times\mathbb R_+^\tau)=\mu_\theta. \)
Consequently, these essential infima always concern populations of positive mass.
Each valuation coordinate has the correct population mean. For every \(\theta\),
\[ \displaystyle \int z_\theta\,\mathrm d\nu_t=V_\theta(t). \tag{1} \]
To justify the unbounded coordinate function, first apply the equation to \(f_K(\phi,z)=\min\{z_\theta,K\}\). Under a jump, its increment lies between zero and \(v_\theta(j(s))\), hence between zero and one. As \(K\) increases, that increment converges to \(v_\theta(j(s))\). Monotone convergence on the left and dominated convergence against \(\mathrm ds\,\mathrm dr_s\) on the right prove (1). In particular, all these means are finite at finite times.
Essential minimum utilities are nondecreasing. For any real \(c\), the indicator of
\( \displaystyle \{(\theta,z):z_\theta<c\} \)
cannot increase under a jump: individuals retain their type and all valuation coordinates increase weakly. Its population mass is therefore nonincreasing. This proves that \(a_\theta(t)\) is nondecreasing. It is finite, since (1) and the positive type mass give
\( \displaystyle a_\theta(t)\le V_\theta(t)/\mu_\theta. \)
Every routed recipient values the current item positively. Some supported type values each item positively, and its mass remains positive. Thus the essential maximum greedy score is strictly positive, possibly infinite. A recipient with zero value has score zero under the stipulated conventions, so cannot attain that maximum. Accordingly, a routed recipient’s own value \(y\) satisfies
\[ \displaystyle \varepsilon\le y\le1. \tag{2} \]
Fix a terminal time \(T\), an evaluating type \(\theta\), and a positive number \(A\) satisfying \(A\ge a_\theta(T)\).
Consider any time \(s\le T\) when the current item has value \(x=v_\theta(j(s))>0\) for \(\theta\). Because \(a_\theta(s)\le A\), the essential maximum greedy score is at least \(x/A\). Indeed, for every positive \(\delta\), a positive mass of type-\(\theta\) individuals has utility below \(A+\delta\); their scores exceed \(x/(A+\delta)\). Letting \(\delta\) decrease to zero gives the assertion.
Suppose the routed recipient has type \(\phi\), own pre-jump utility \(b=z_\phi\), and own item value \(y=v_\phi(j(s))\). Greediness implies, when \(b>0\),
\[ \displaystyle \frac{y}{b}\ge\frac{x}{A},\qquad b\le\frac{Ay}{x}. \tag{3} \]
The second inequality also holds when \(b=0\). Moreover,
\[ \displaystyle \frac{x}{y}\le\frac1\varepsilon,\qquad b\le\frac A\varepsilon. \tag{4} \]
Define a nonnegative function on the nonnegative real line by
\( \displaystyle h_A(u)= \begin{cases} 1/\varepsilon,&0\le u\le1,\\ \min\{1/\varepsilon,A/(u-1)\},&1<u\le1+A/\varepsilon,\\ 0,&u>1+A/\varepsilon, \end{cases} \qquad F_A(b)=\int_0^b h_A(u)\,\mathrm du. \)
We claim that every routed jump satisfies
\[ \displaystyle F_A(b+y)-F_A(b)\ge x. \tag{5} \]
If \(x=0\), this follows from the monotonicity of \(F_A\). If \(x>0\), take \(u\) in the interval from \(b\) to \(b+y\). By (2) and (4),
\( \displaystyle u\le b+1\le A/\varepsilon+1. \)
For \(u\le1\), we have \(h_A(u)=1/\varepsilon\ge x/y\). For \(u>1\), we have \(u-1\le b\). If \(b>0\), (3) gives
\( \displaystyle \frac{A}{u-1}\ge\frac Ab\ge\frac xy. \)
Together with \(1/\varepsilon\ge x/y\), this again implies \(h_A(u)\ge x/y\). When \(b=0\), the entire interval lies below one, so the first case suffices. Integrating proves (5).
The total potential is
\[ \displaystyle \sup_{b\ge0}F_A(b) =\frac1\varepsilon+ \int_0^{A/\varepsilon}\min\{1/\varepsilon,A/u\}\,\mathrm du =\frac1\varepsilon+C_\varepsilon A. \tag{6} \]
For the last equality, split the integral at \(A\varepsilon\). The first part equals \(A\), and the second equals
\( \displaystyle A\log\!\left(\frac{A/\varepsilon}{A\varepsilon}\right) =2A\log(1/\varepsilon). \)
Fix a recipient type \(\phi\). The inequality
\[ \displaystyle z_\theta\le F_A(z_\phi) \tag{7} \]
holds initially. It is preserved by every routed jump: the left side increases by \(x\), while the right side increases by at least \(x\), by (5).
Here is a direct measure-level justification. Let
\( \displaystyle B=\{(\phi,z):z_\theta>F_A(z_\phi)\}. \)
No routed jump can enter \(B\) from its complement. Therefore the increment of the indicator of \(B\) is everywhere nonpositive on routed states. The forward equation gives
\( \displaystyle \nu_t(B)\le\nu_0(B)=0\qquad(0\le t\le T). \)
Thus (7) holds almost everywhere, without needing individual lineages or a particle representation.
Applying (6), and then taking \(A\) down to \(a_\theta(T)\), gives, for every recipient type \(\phi\),
\( \displaystyle
z_\theta\le C_\varepsilon a_\theta(T)+\frac1\varepsilon
\quad\text{for $\nu_T\)-almost every \((\phi,z)\)}. \tag{8}
\)
If \(a_\theta(T)=0\), use a countable sequence of positive \(A\) decreasing to zero; the same conclusion follows.
Integrating (8) over the whole probability measure and using (1) yields
\[ \displaystyle V_\theta(T)\le C_\varepsilon a_\theta(T)+\frac1\varepsilon, \qquad a_\theta(T)\ge \frac{V_\theta(T)-1/\varepsilon}{C_\varepsilon}. \tag{9} \]
Admissibility requires \(V_\theta(T)\) to diverge for every supported type. Since the support is finite, (9) implies
\[ \displaystyle \min_{\theta\in\Theta}a_\theta(T)\longrightarrow\infty. \tag{10} \]
Once these minima are positive, (8) bounds every envy ratio:
\[ \displaystyle E(T)\le C_\varepsilon+ \frac{1}{\varepsilon\min_\theta a_\theta(T)}. \tag{11} \]
Consequently, every admissible greedy trajectory satisfies
\[ \displaystyle \limsup_{T\to\infty}E(T)\le C_\varepsilon. \tag{12} \]
This also supplies an explicit population-independent error bound. If \(V_{\min}(T)=\min_\theta V_\theta(T)>1/\varepsilon\), then
\[ \displaystyle E(T)\le C_\varepsilon+ \frac{C_\varepsilon}{\varepsilon V_{\min}(T)-1}. \tag{13} \]
For the lower bound, use two types, called \(A\) and \(B\), each of mass one half.
Suppose initially each type is concentrated at one bundle state. Consider an item for which type \(A\)’s state attains the maximum greedy score. Let its pre-jump state be \(z\), and let the item vector be \(\mathbf v\).
During an interval of length one half, move type-\(A\) mass according to
\( \displaystyle \nu_{t+s}\big|_A =(1/2-s)\delta_{(A,z)} +s\delta_{(A,z+\mathbf v)}, \qquad 0\le s\le1/2, \)
leaving the type-\(B\) population fixed. For \(s<1/2\), choose
\( \displaystyle r_{t+s}=\delta_{(A,z)}. \)
This routing measure has mass one and is absolutely continuous with respect to the current population measure. Its receiving state remains a maximum-score state: its score is unchanged, the other type’s score is unchanged, and the already-served portion of type \(A\) has a smaller score because its own utility has increased.
The measure formula satisfies the forward equation for every bounded test function. At the end, all type-\(A\) individuals have received one indivisible item and are again synchronized.
The same construction works for type \(B\). Hence any integral greedy sequence for two equal-weight agents can be realized exactly at synchronized times, with each allocation taking one half unit of society time. Each individual bundle changes only by the stipulated full item jump.
Assume first that \(\varepsilon<1\). Choose dyadic rationals
\( \displaystyle \varepsilon<q=p_0<p_1<\cdots<p_k=1. \)
Use the following fixed menu, writing the two entries as the values of types \(A\) and \(B\):
There are \(2k+2\) kinds. Every positive value is binary-rational and lies in the required interval. The types are distinct, and every kind has a positive supported valuation.
Define
\[ \displaystyle S=\sum_{\ell=1}^k\left(1-\frac{p_{\ell-1}}{p_\ell}\right), \qquad D=1+2S. \tag{14} \]
We construct one infinite stream whose envy has limit superior at least \(D\).
At a synchronized block boundary, let:
Type \(B\) will receive only private items, so \(R_{\mathrm{old}}\) is an integer.
Choose a sufficiently large integer \(R>R_{\mathrm{old}}\).
First give type \(B\) exactly \(R-R_{\mathrm{old}}\) private items. These allocations are greedy because type \(A\)’s value is zero. Type \(B\)’s own utility becomes \(R\).
Next, for indices \(\ell=0,\ldots,k\), supply the item \((p_\ell,1)\) until type \(A\)’s own utility first reaches or exceeds \(p_\ell R\). Every allocation before stopping goes greedily to type \(A\): if its current own utility is \(U<p_\ell R\), then
\( \displaystyle \frac{p_\ell}{U}>\frac1R. \)
The stopping utility belongs to the interval from \(p_\ell R\), inclusive, to \(p_\ell R+p_\ell\), exclusive.
Finally, for indices \(\ell=k-1,\ldots,0\), supply the item \((1,p_\ell)\) until type \(A\)’s own utility first reaches or exceeds \(R/p_\ell\). Again, every allocation goes greedily to type \(A\), since before stopping
\( \displaystyle U<R/p_\ell \quad\Longrightarrow\quad \frac1U>\frac{p_\ell}{R}. \)
The stopping utility belongs to the interval from \(R/p_\ell\), inclusive, to \(R/p_\ell+1\), exclusive.
All phases start below their stopping thresholds if \(R\) is sufficiently large: the fixed menu has strictly separated successive thresholds, while every stopping overshoot is less than one. We also choose \(R\) so that \(H<qR\).
Let \(\Delta W\) be the increase in type \(B\)’s value for a type-\(A\) bundle.
The first shared phase contains
\( \displaystyle \left\lceil\frac{qR-H}{q}\right\rceil =R-\frac Hq+O(1) \)
items, each worth one to type \(B\).
For each subsequent item \((p_\ell,1)\), the number allocated is
\( \displaystyle R\left(1-\frac{p_{\ell-1}}{p_\ell}\right)+O(1). \)
Indeed, the phase begins at own utility \(p_{\ell-1}R\) plus an overshoot less than one, and each item adds \(p_\ell\).
For the descending phase with item \((1,p_\ell)\), the number allocated is
\( \displaystyle R\left(\frac1{p_\ell}-\frac1{p_{\ell+1}}\right)+O(1). \)
Multiplying by its type-\(B\) value \(p_\ell\), its contribution is
\( \displaystyle R\left(1-\frac{p_\ell}{p_{\ell+1}}\right)+O(1). \)
The menu is fixed, so the sum of all rounding errors is bounded independently of \(H\) and \(R\). Thus
\[ \displaystyle \Delta W=DR-\frac Hq+O_{\mathrm{menu}}(1). \tag{15} \]
At the block endpoint, type \(A\)’s own utility satisfies
\[ \displaystyle R/q\le H_{\mathrm{new}}<R/q+1. \tag{16} \]
Repeat these blocks. In block \(h\), choose an integer \(R_h\) large enough for all threshold conditions above and satisfying
\( \displaystyle R_h\ge h\bigl(1+H_{h-1}+W_{h-1}+R_{h-1}\bigr). \)
It follows that both \(H_{h-1}/R_h\) and \(W_{h-1}/R_h\) tend to zero. Equation (15) therefore gives
\[ \displaystyle \frac{W_h}{R_h} =D+\frac{W_{h-1}-H_{h-1}/q}{R_h} +O_{\mathrm{menu}}(1/R_h) \longrightarrow D. \tag{17} \]
Realize every allocation using Section 4. The resulting stream uses the same finite menu throughout. Every block has finitely many phases, and the private phase durations grow without bound; hence there are only finitely many phases on each bounded time interval.
Let \(T_h\) be the synchronized endpoint of block \(h\). At that time, type \(B\) has own utility \(R_h\) and values every type-\(A\) bundle at \(W_h\). Consequently,
\[ \displaystyle E(T_h)\ge W_h/R_h\longrightarrow D. \tag{18} \]
The stream is admissible. By (1), synchronization, and the fact that type \(B\)’s private bundles have zero value to type \(A\),
\( \displaystyle V_A(T_h)=H_h/2\ge R_h/(2q), \qquad V_B(T_h)=(R_h+W_h)/2\ge R_h/2. \)
Both diverge; their monotonicity gives the required divergence along the entire stream.
The sum in (14) is the right-endpoint Riemann sum
\( \displaystyle S=\sum_{\ell=1}^k\frac{p_\ell-p_{\ell-1}}{p_\ell} \)
for the integral of \(1/u\) over the interval from \(q\) to one. Dyadic partitions with mesh tending to zero therefore give
\( \displaystyle S\longrightarrow\int_q^1\frac{\mathrm du}{u}=\log(1/q). \)
For every positive \(\eta\), first choose a dyadic \(q>\varepsilon\) sufficiently close to \(\varepsilon\), and then choose a sufficiently fine dyadic partition. These choices ensure
\( \displaystyle D>1+2\log(1/\varepsilon)-\eta. \)
Sections 4–5 then give a single admissible infinite stream in a finite supported society with a fixed finite menu satisfying
\( \displaystyle \limsup_{t\to\infty}E(t)> 1+2\log(1/\varepsilon)-\eta. \)
Taking the supremum and then letting \(\eta\) decrease to zero proves the lower bound.
For \(\varepsilon=1\), use one type of mass one and one item valued at one. During the interval from integer time \(k\) to \(k+1\), move mass continuously from utility \(k\) to utility \(k+1\), routing to the remaining utility-\(k\) population. This is greedy and satisfies the jump equation. At interior times with \(k\) positive, its envy is \((k+1)/k\), so its asymptotic factor is one.
Together with (12), this establishes the sharp value for every allowed \(\varepsilon\).
⟨structured verdict parsed — rendered above⟩
The prover’s claim is proved. The argument establishes the sharp value \(E^*(\varepsilon)=1+2\log(1/\varepsilon)\) directly for the stipulated bundle-state measure dynamics. I found no mathematical errors or gaps requiring clarification.
The prover explicitly uses the time-integrated forward equation for bounded measurable test functions. This is the natural interpretation of the stated dynamics, including at stream phase boundaries. The subsequent indicator arguments and piecewise linear constructions consistently use this interpretation.
Type indicators have zero increment under every bundle jump, so each type retains its positive mass \(\mu_\theta\).
The truncation argument proving equation (1) is valid. The functions \(\min\{z_\theta,K\}\) increase to \(z_\theta\), while their jump increments lie between zero and \(v_\theta(j(s))\le1\) and converge pointwise to that value. The time-routing measure has finite mass on every bounded time interval, so the stated dominated convergence argument applies. Consequently, the coordinate means are finite and equal \(V_\theta(t)\).
The sublevel-set indicators used in Section 1 cannot increase under a jump. Their masses are therefore nonincreasing, which establishes monotonicity of \(a_\theta(t)\).
The bound \(a_\theta(t)\le V_\theta(t)/\mu_\theta\) follows from the positive type mass and the population coordinate mean. In particular, every terminal essential minimum used in the potential argument is finite.
Every item is positively valued by some supported type of positive mass. At every state of that type, its score is positive or infinite because its own utility is a finite nonnegative coordinate. Thus the essential maximum score is strictly positive.
A state with zero own value has score zero under both applicable denominator conventions. Such a state cannot be routed. Therefore equation (2), including its lower bound \(\varepsilon\), is justified.
For fixed \(A>0\) with \(A\ge a_\theta(T)\), monotonicity gives \(a_\theta(s)\le A\) throughout the relevant interval. For each \(\delta>0\), positive type-\(\theta\) mass has own utility below \(A+\delta\). When \(x>0\), this establishes the asserted lower bound \(x/A\) on the essential maximum score.
Since routing is concentrated on maximizing states, equation (3) holds routing-almost everywhere when \(b>0\); its bound on \(b\) also holds when \(b=0\). Equation (4) correctly uses \(x\ge\varepsilon\), \(x\le1\), and \(y\le1\).
The proof of equation (5) checks the entire jump interval, rather than treating an indivisible item as an infinitesimal utility increment. The relevant inequalities are correct: \(b+y\le b+1\le A/\varepsilon+1\), and for \(u>1\), \(u-1\le b\). Both branches defining \(h_A\) therefore provide the required lower bound \(x/y\). The case \(b=0\) is handled separately and correctly.
The calculation in equation (6) is also correct. The breakpoint is \(A\varepsilon\), which lies within \([0,A/\varepsilon]\) because \(\varepsilon\le1\). The integral contributes \(A+2A\log(1/\varepsilon)\), in addition to the initial contribution \(1/\varepsilon\). This calculation remains valid at \(\varepsilon=1\).
The bad set in Section 3 is a fixed measurable set once \(A\), \(\theta\), and \(\phi\) are fixed. Equation (5) prevents routed jumps from entering it from its complement. Jumps starting inside it have indicator increment at most zero, and jumps of other recipient types leave its indicator unchanged.
The forward equation therefore shows that its initially zero mass remains zero. This is a valid measure-level invariance argument and requires neither individual lineages nor an unstated particle representation.
Taking \(A\) down to \(a_\theta(T)\) is legitimate using a countable sequence and continuity of the numerical bound. The prover explicitly covers \(a_\theta(T)=0\). Because the support is finite, the resulting almost-everywhere statements can be combined over all types.
Integrating equation (8) gives equation (9). Admissibility then forces every \(a_\theta(T)\) to diverge. Finiteness of the support is essential to equation (10) and is available in the task.
Once all own-utility minima are positive, equation (8) bounds the numerator of every relevant envy ratio, while the denominator is bounded below by the corresponding type minimum. This establishes equation (11) for the product-measure essential supremum.
The substitution yielding equation (13) is correct, including the condition \(V_{\min}(T)>1/\varepsilon\). Thus the upper bound applies to every admissible greedy trajectory in the question.
The construction in Section 4 satisfies all required routing conditions during each allocation interval. For \(s<1/2\), the pre-jump atom has positive population mass, so routing a unit Dirac measure there is absolutely continuous with respect to the population measure. The task imposes no upper bound on the routing density.
The receiving atom retains its original score; the other type retains its score; and the served portion cannot have a larger score. Hence routing remains greedy. The derivative of the displayed population formula equals the specified full-jump increment for every bounded test function.
Concatenating these intervals yields the stated synchronized allocations in the time-integrated sense. Endpoint changes cause no difficulty.
The menu has exactly \(2k+2\) kinds. All positive values are dyadic rationals in \([\varepsilon,1]\), both type masses are admissible, and the private item makes the two valuation types distinct.
Private allocations are greedy even if type \(A\) has zero utility, because its score is \(0/0=0\). During shared phases, the displayed strict score comparisons correctly select type \(A\).
For the fixed partition, consecutive stopping thresholds are strictly separated by quantities proportional to \(R\). The overshoots are less than one. Thus the stated sufficiently large choice of \(R\) makes every phase start below its threshold, including the initial requirement \(H<qR\).
The phase counts and contributions in equation (15) are correct. Errors caused by overshoots and integer rounding are bounded by constants depending only on the fixed menu. In particular, those bounds are independent of \(H\) and \(R\).
The descending contributions reproduce the same sum \(S\) as the ascending contributions, giving \(D=1+2S\). The final threshold also gives equation (16).
The growth condition on \(R_h\) makes the previous block’s utilities negligible relative to \(R_h\), validating equation (17). It also forces the private-phase durations to diverge, excluding accumulation of infinitely many phases in bounded time.
At synchronized endpoints, the relevant cross-type pairs have positive product-measure mass, so equation (18) is a valid lower bound on \(E(T_h)\). The displayed monopolistic-utility identities follow from synchronization and equation (1), and prove admissibility for both types. The routing choices use current states and items, satisfying the information restriction.
The sum \(S\) is indeed a right-endpoint Riemann sum for \(1/u\). Dyadic partitions can approach the integral for fixed dyadic \(q> \varepsilon\), and such \(q\) can approach \(\varepsilon\) when \(\varepsilon<1\).
The quantifier order is correct: each approximation uses its own finite menu and one admissible infinite stream. The supremum defining \(E^*(\varepsilon)\) permits this; attainment by one menu is not required.
The one-type construction for \(\varepsilon=1\) satisfies the dynamics and has asymptotic envy one. Its initial zero-utility behavior does not affect the limiting claim.
The cited finite-agent bounds occur in [Section 7 and Appendix D of the referenced paper](https://arxiv.org/pdf/2408.02403). They serve as background here; the proof does not apply them to the continuous society or depend on an unverified transfer theorem.
No repair feedback is necessary. The output resolves the sharp-factor question but makes no computational complexity classification, so the required continuous_status is "unresolved" under the supplied schema.
⟨structured verdict parsed — rendered above⟩
Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.