Answer: Such a fully decentralized, uncoordinated policy exists. It guarantees \(\mathbb E R_\infty(T)\le C(K,\rho_{\min},\Delta)\log(1+T)\), with a finite coefficient depending only on \(K\) and \(\Delta\).
Theorem 3 — class: algorithmic
PDF text-layer excerpt — equations and symbol order may be approximateAssume that each man follows Algorithm 1 with ηt = 1 √ t and γt = M log t t , where M = 4n c log T and c = 1 8 mink∈[n]{∆, µmkwk}.
statement extracted from the paper’s text layer; consult the paper for the exact wording
Given finite man and woman cohorts \(A,B\) with rational masses ρ,σ, type-level strict priorities, bounded unknown reward laws \(D_{ab}\), and the hierarchical diagonal condition, define a nonatomic capacity-clearing process: at round \(t\), each infinitesimal man uses only his private history to choose a woman type, proposal mass \(p^t_{ab}\) is accepted up to capacity σ_b in priority order, producing accepted flow \(q^t_{ab}\), while accepted agents receive samples and rejected agents receive only rejection. Does there exist a fully decentralized policy with \(\mathbb{E}R_∞(T)=O(\log T)\), where \(R_∞(T)=\sum_{t=1}^{T}(1-\sum_{k=1}^{K}q^t_{a_kb_k})\), with the coefficient controlled by \(K\), \(\min_kρ_{a_k}\), and the type-level gap Δ?
Let \(A,B\) each contain \(K\ge2\) types of nonatomic men and women. A society is given by its support: the list of the \(\tau\) types with positive mass, each mass a binary rational; \(\tau\) denotes support size, not the number of possible types. Here \(m=K\), \(\tau=2K\), and \(L\) is the bit length of the masses and priority lists; state computational costs in \(m,\tau,L,T\) and any additional parameters.
The masses satisfy \(\sum_a\rho_a=\sum_b\sigma_b=1\). Each woman type \(b\) has a strict priority order \(\succ_b\) over man types. Each pair has an unknown reward law \(D_{ab}\) supported on \([0,1]\), with mean \(\mu_{ab}>0\); each man's means are distinct, and \(\Delta=\min_{a,b\ne b'}|\mu_{ab}-\mu_{ab'}|>0\).
There exist orderings \(A=(a_1,\ldots,a_K)\), \(B=(b_1,\ldots,b_K)\) satisfying \(\rho_{a_k}=\sigma_{b_k}\), \(\mu_{a_kb_k}>\mu_{a_kb_j}\), and \(a_k\succ_{b_k}a_j\) whenever \(j>k\). These witnessing orderings are not disclosed to learners.
Every round, each man proposes to exactly one woman type. If \(p^t_{ab}\) is proposal mass, accepted mass is \(q^t_{ab}=\min\{p^t_{ab},(\sigma_b-\sum_{a'\succ_b a}p^t_{a'b})_+\}\); capacities reset each round.
Within a proposing type, rationing is uniform and independent of private history: each proposer is accepted with probability \(q^t_{ab}/p^t_{ab}\) when \(p^t_{ab}>0\). Accepted men observe fresh private samples from \(D_{ab}\); rejected men observe only rejection. Population histories and flows follow the corresponding probability-law evolution, without finite-population sampling fluctuations.
A fully decentralized, uncoordinated policy uses only a man's type, his own past proposals and feedback, and private randomness. It may know the support, masses, round and horizon, but not reward laws, means, priorities, witnessing orderings, aggregate flows or others' histories; no communication, shared randomness or coordinated exploration phases are allowed.
Define \(\rho_{\min}=\min_a\rho_a\) and \(R_\infty(T)=\sum_{t=1}^T(1-\sum_{k=1}^Kq^t_{a_kb_k})\).
Does there exist such a policy guaranteeing \(\mathbb E R_\infty(T)\le C(K,\rho_{\min},\Delta)\log(1+T)\) for every admissible instance and horizon, with finite \(C\) independent of all other instance details, or is this guarantee impossible?
For finite hierarchical markets, decentralized EXP has \(R(T)=\widetilde O(n^3c^{-(n+2)}\log T+n^2c^{-1}\log^3T)\), where \(c=\frac18\min_k\{\Delta,\mu_{m_kw_k}\}\); the hidden factors involve \(\log\log T\), and regret counts rounds with any incorrect proposal. [Source](https://arxiv.org/html/2407.21294v2).
No result settling this precise nonatomic private-feedback model was located in the literature search.
A fully decentralized, uncoordinated policy attains the requested guarantee. In fact, its coefficient need not depend on \(\rho_{\min}\).
One explicit bound is
\(\displaystyle \mathbb E R_\infty(T) \le \frac{2500K^2}{\Delta^2} \left(1+\frac{8}{\Delta}\right)^{K-1} \log(1+T). \)
The coefficient is large, but finite and uniform over all the other instance details. The policy does not know \(\Delta\), the priorities, or the witnessing orderings.
Let \(\mathcal P_m=\{w\in\mathbb R_{\ge0}^m:\sum_b w_b=1\}\), where \(m=K\).
Each man independently maintains a vector of cumulative estimated losses \(\widehat S_t\), initially \(\widehat S_0=0\). In round \(t\), he computes
\(\displaystyle w_t=\operatorname*{arg\,min}_{w\in\mathcal P_m} \left\{ \langle \widehat S_{t-1},w\rangle +16\sqrt t\left(1-\sum_b\sqrt{w_b}\right) \right\}. \)
He proposes to a woman type \(I_t\) drawn from \(w_t\), using private randomness.
His observed loss is
\(\displaystyle \ell_{t,I_t}= \begin{cases} (1-X_t)/2,&\text{if accepted and observing reward }X_t,\\ 1,&\text{if rejected}. \end{cases} \)
He updates using importance weighting:
\(\displaystyle \widehat\ell_{t,b} =\frac{\mathbf1\{I_t=b\}\ell_{t,I_t}}{w_{t,b}}, \qquad \widehat S_t=\widehat S_{t-1}+\widehat\ell_t. \)
The minimizer exists, is unique, and has strictly positive coordinates. Existence follows from compactness and continuity; uniqueness follows from strict convexity. To verify positivity, transferring a sufficiently small mass \(\varepsilon\) from a positive coordinate to a zero coordinate changes the objective by \(-16\sqrt{t\varepsilon}+O(\varepsilon)\), which is negative. Thus every update is well-defined.
Equivalently, the policy assigns gain \((1+X_t)/2\) to acceptance and gain zero to rejection. This transformation preserves reward-mean differences between accepted options while giving every accepted option expected gain at least \(1/2\).
Every man runs this rule using only his private history and the current round. There are no exploration phases, communication, shared randomness, or aggregate observations.
We first establish the bandit estimate needed below.
Consider the preceding policy in an \(m\)-armed bandit environment with losses in \([0,1]\), selected without observing the learner’s current sampled action. Write
\(\displaystyle u_{t,b}=\mathbb E w_{t,b}, \qquad U_j(T)=\mathbb E\sum_{t=1}^T (\ell_{t,I_t}-\ell_{t,j}). \)
For every fixed arm \(j\),
\[\displaystyle U_j(T) \le 17\sum_{t=1}^T\sum_{b\ne j}\sqrt{\frac{u_{t,b}}t} +16\sqrt m. \tag{1} \]
In particular, \(j\) need not be optimal in hindsight.
The external ingredient is Lemma 11, part 4, of [Zimmert and Seldin, *Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits*](https://jmlr.org/papers/volume22/19-753/19-753.pdf). For importance-weighted estimators, Tsallis parameter \(1/2\), symmetric regularization, and learning rate \(\eta_t\le1/4\), its instantaneous stability bound, for any arm \(j\), is
\[\displaystyle \mathbb E\!\left[ \ell_{t,I_t}+ \Phi_t(-\widehat S_t)-\Phi_t(-\widehat S_{t-1}) \right] \le \frac{\eta_t}{2} \sum_{b\ne j}\left(\sqrt{u_{t,b}}+3u_{t,b}\right). \tag{2} \]
Here \(\Phi_t\) is the conjugate potential of the scaled regularizer.
Our regularizer agrees, up to an additive constant, with that lemma’s regularizer for \(\eta_t=1/(4\sqrt t)\). The estimators are importance-weighted, losses lie in \([0,1]\), and \(\eta_t\le1/4\) in every round. Thus all its hypotheses hold. Since \(u_{t,b}\le\sqrt{u_{t,b}}\), summing (2) gives stability at most
\[\displaystyle \frac12\sum_{t=1}^T\sum_{b\ne j} \sqrt{\frac{u_{t,b}}t}. \tag{3} \]
For completeness, the remaining calculation does not require a self-bounding bandit theorem.
Put
\(\displaystyle F_t(w)=16\sqrt t\left(1-\sum_b\sqrt{w_b}\right), \qquad \Phi_t(y)=\max_{w\in\mathcal P_m} \{\langle y,w\rangle-F_t(w)\}. \)
Adding and subtracting potential differences decomposes \(U_j(T)\) into stability and
\(\displaystyle P_j= \mathbb E\sum_{t=1}^T \left[ \Phi_t(-\widehat S_{t-1}) -\Phi_t(-\widehat S_t)-\ell_{t,j} \right]. \)
The estimators are unbiased. Also, \(F_T(e_j)=0\), so
\(\Phi_T(-\widehat S_T)\ge-\widehat S_{T,j}\).
Consequently, telescoping yields
\(\displaystyle P_j\le \Phi_1(0)+ \sum_{t=2}^T \mathbb E\left[ \Phi_t(-\widehat S_{t-1}) -\Phi_{t-1}(-\widehat S_{t-1}) \right]. \)
Using \(w_t\) as a feasible point in the maximization defining \(\Phi_{t-1}\), each potential difference is at most
\(\displaystyle 16(\sqrt t-\sqrt{t-1}) \left(\sum_b\sqrt{w_{t,b}}-1\right). \)
Now \(\Phi_1(0)=16(\sqrt m-1)\), and, for any \(j\),
\(\displaystyle \sum_b\sqrt{w_{t,b}}-1 \le\sum_{b\ne j}\sqrt{w_{t,b}}. \)
Jensen’s inequality and
\(\sqrt t-\sqrt{t-1}\le1/\sqrt t\) therefore give
\(\displaystyle P_j\le16\sqrt m+ 16\sum_{t=2}^T\sum_{b\ne j} \sqrt{\frac{u_{t,b}}t}. \)
Combining this with (3) proves (1).
Let \(H_T=\sum_{t=1}^T1/t\) and
\(z_j(T)=\sum_{t=1}^T(1-u_{t,j})\).
Cauchy–Schwarz also gives the convenient consequence
\[\displaystyle U_j(T)\le 17\sqrt{(m-1)H_Tz_j(T)}+16\sqrt m. \tag{4} \]
Fix any witnessing orderings for the analysis, and abbreviate
\(\rho_k=\rho_{a_k}=\sigma_{b_k}\).
Let
\(\displaystyle x_{k,b}^t=\Pr(I_t=b\mid\text{man type }a_k), \qquad p_{a_kb}^t=\rho_kx_{k,b}^t. \)
These probabilities average over the type’s private histories and current randomization. Define its incorrect proposal mass by
\(\displaystyle e_k^t=\rho_k-p_{a_kb_k}^t =\rho_k(1-x_{k,b_k}^t). \)
The stipulated probability-law population evolution makes these flows deterministic. A single nonatomic man cannot change them.
Write \(f_{k,b}^t=q_{a_kb}^t/p_{a_kb}^t\) for his acceptance probability. The policy gives every arm positive proposal mass, so this ratio is defined. History-independent rationing and fresh reward samples imply that, conditional on his private history and proposal to \(b\), his expected transformed gain is
\(\displaystyle v_{k,b}^t=\nu_{k,b}f_{k,b}^t, \qquad \nu_{k,b}=\frac{1+\mu_{a_kb}}2. \)
Thus a representative man encounters a valid bandit environment: each arm’s feedback law in round \(t\) is independent of his private history and current action randomization.
Formally, one can construct an unobserved loss vector by drawing, for every arm, a potential acceptance indicator with probability \(f_{k,b}^t\) and a fresh reward sample. Revealing only its selected coordinate reproduces the prescribed private-feedback law. This construction is solely for the analysis.
Define
\(\displaystyle s_{k-1}^t=\sum_{i<k}e_i^t, \qquad g=\frac{\Delta}{2}. \)
We prove the following inequality for the expected loss difference against the intended partner:
\[\displaystyle v_{k,b_k}^t-\sum_bx_{k,b}^tv_{k,b}^t \ge g(1-x_{k,b_k}^t)-\frac{2s_{k-1}^t}{\rho_k}. \tag{5} \]
Every type ranked above \(a_k\) at \(b_k\) must be earlier in the witness. For \(i<k\), its proposal mass to \(b_k\) is at most \(e_i^t\), because \(b_k\ne b_i\).
Hence total higher-priority proposal mass at \(b_k\) is at most \(s_{k-1}^t\). Since \(p_{a_kb_k}^t\le\rho_k=\sigma_{b_k}\), the rationing formula implies
\(\displaystyle f_{k,b_k}^t\ge1-\frac{s_{k-1}^t}{\rho_k}. \)
Indeed, when higher-priority mass \(h\le\rho_k\), the acceptance ratio is at least
\((\rho_k-h)/\rho_k\); when \(h>\rho_k\), the displayed lower bound is nonpositive.
Because \(\nu_{k,b_k}\le1\), it follows that
\[\displaystyle v_{k,b_k}^t \ge\nu_{k,b_k}-\frac{s_{k-1}^t}{\rho_k}. \tag{6} \]
For \(i<k\), woman type \(b_i\) ranks \(a_i\) above \(a_k\). Therefore
\(\displaystyle q_{a_kb_i}^t \le \sigma_{b_i}-p_{a_ib_i}^t =\rho_i-p_{a_ib_i}^t=e_i^t. \)
As transformed gains are at most one,
\[\displaystyle \sum_{i<k}x_{k,b_i}^tv_{k,b_i}^t =\frac1{\rho_k}\sum_{i<k}\nu_{k,b_i}q_{a_kb_i}^t \le\frac{s_{k-1}^t}{\rho_k}. \tag{7} \]
For \(i>k\), the witnessing preference inequality and the definition of \(\Delta\) imply
\(\displaystyle \nu_{k,b_k}-\nu_{k,b_i}\ge g. \)
Also \(v_{k,b_i}^t\le\nu_{k,b_i}\).
Combining these observations with (6)–(7), the left-hand side of (5) is at least
\(\displaystyle \nu_{k,b_k}(1-x_{k,b_k}^t) -\sum_{i>k}\nu_{k,b_i}x_{k,b_i}^t -\frac{2s_{k-1}^t}{\rho_k}. \)
For a later partner, its coefficient is at least \(g\). For an earlier partner, its coefficient is \(\nu_{k,b_k}\ge1/2\ge g\), since \(\Delta\le1\). This proves (5).
Notice that this argument bounds accepted mass at earlier partners. Their individual acceptance probabilities can be large when very little mass proposes to them; no uniform pointwise gap for those arms is assumed.
Let
\(\displaystyle E_k(T)=\sum_{t=1}^Te_k^t, \qquad P_k(T)=\sum_{i=1}^kE_i(T), \qquad P_0(T)=0. \)
For a representative man of type \(a_k\), the expected number of incorrect proposals is
\(\displaystyle z_k(T)=\frac{E_k(T)}{\rho_k}. \)
Let \(U_k(T)\) denote his expected loss difference against always selecting \(b_k\), evaluated in his actual bandit environment. Summing (5) gives
\(\displaystyle gz_k(T) \le U_k(T)+\frac{2P_{k-1}(T)}{\rho_k}. \)
Applying (4) with comparison arm \(b_k\),
\(\displaystyle gz_k(T) \le 17\sqrt{(m-1)H_Tz_k(T)} +16\sqrt m+\frac{2P_{k-1}(T)}{\rho_k}. \)
Young’s inequality gives
\(\displaystyle 17\sqrt{(m-1)H_Tz} \le\frac g2z+\frac{289(m-1)H_T}{2g}. \)
Consequently,
\[\displaystyle E_k(T)\le \rho_k A_T+\frac4gP_{k-1}(T), \qquad A_T=\frac{289(m-1)H_T}{g^2} +\frac{32\sqrt m}{g}. \tag{8} \]
This is the required hierarchical recursion. In particular, multiplying by \(\rho_k\) cancels the inverse mass in the disturbance term.
Writing \(r=1+4/g\), equation (8) implies
\(\displaystyle P_k(T)\le rP_{k-1}(T)+\rho_kA_T. \)
Induction and \(\sum_k\rho_k=1\) yield
\[\displaystyle P_K(T)\le A_T\sum_{k=1}^K\rho_kr^{K-k} \le A_Tr^{K-1}. \tag{9} \]
It remains to account for correct proposals that are rejected. The same capacity argument used above gives
\(\displaystyle q_{a_kb_k}^t\ge p_{a_kb_k}^t-s_{k-1}^t. \)
Thus
\(\displaystyle \rho_k-q_{a_kb_k}^t\le e_k^t+s_{k-1}^t. \)
Since the diagonal masses sum to one,
\[\displaystyle R_\infty(T) \le \sum_{k=1}^K(K-k+1)E_k(T) \le KP_K(T). \tag{10} \]
Finally, for integer \(T\ge1\),
\(H_T\le1+\log T\le2\log(1+T)\) and
\(1\le2\log(1+T)\).
Using \(g=\Delta/2\), \(\Delta\le1\), and \(\sqrt m\le m\), we obtain
\(\displaystyle A_T \le \left( \frac{2312(m-1)}{\Delta^2} +\frac{128\sqrt m}{\Delta} \right)\log(1+T) \le \frac{2500m}{\Delta^2}\log(1+T). \)
Combining (9)–(10) and \(m=K\) proves the announced bound.
Under the stipulated deterministic probability-law evolution, \(R_\infty(T)\) itself is deterministic. The expectation in the question therefore satisfies the same bound.
Costs below are per man; a nonatomic population does not have a finite total number of learners.
The policy stores \(O(m)\) real-valued scores. It needs only the woman-type labels and round number. Parsing the public support takes at most \(O(L+\tau)\) work; no expansion of rational masses into individual agents is required.
Each probability update reduces to one scalar monotone equation. Given scores \(s_b\), put \(d_b=s_b-\min_c s_c\). The optimal probabilities are
\(\displaystyle w_{t,b}=\frac{64t}{(d_b+z)^2}, \)
where \(z\) is the unique solution of
\(\displaystyle \sum_b\frac{64t}{(d_b+z)^2}=1, \qquad 8\sqrt t\le z\le8\sqrt{mt}. \)
Thus, in an exact-real model with scalar-normalization cost \(Q\), the per-man horizon cost is \(O(T(m+Q))\), with \(O(m)\) real registers. For numerical evaluation, bisection computes these probabilities to total absolute error at most \(\varepsilon\) in \(O(m\log(m/\varepsilon))\) arithmetic operations per round. The proved guarantee concerns the exact policy.
A bit-operation bound requires an additional representation or access model for reward samples, since the problem permits arbitrary real-valued samples. Likewise, no finite explicit representation of the entire population-history law is assumed or needed by the learners.
The conclusion uses the stated nonatomic, history-independent rationing model. It follows from the bandit stability estimate and the mass inequalities above, rather than from transferring the cited finite-market EXP guarantee.
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Verdict: proved. The proposed policy satisfies the information restrictions, and the argument establishes the announced logarithmic regret bound. I found no major or minor mathematical issues.
The computational claims are explicitly restricted to arithmetic and exact-real models. They do not establish a standard bit-complexity classification for arbitrary real-valued reward samples.
The objective is continuous on the compact probability simplex and strictly convex, so its minimizer exists and is unique. The stated perturbation at a zero coordinate has a negative term of order \(\sqrt{\varepsilon}\), dominating the terms of order \(\varepsilon\). Thus all coordinates are positive.
Starting from finite scores, positivity makes each importance-weighted update finite. This validates the policy recursively at every finite round.
Accepted losses belong to \([0,1/2]\), and rejected losses equal \(1\). Consequently, all observed losses lie in \([0,1]\), as required by the bandit estimate.
The policy uses only private observations, private randomness, woman-type labels, and the round number. A common deterministic update rule and knowledge of the round do not introduce communication, shared randomness, or coordinated exploration phases.
I checked the cited result against the original [Zimmert–Seldin paper, Lemma 11, part 4](https://jmlr.org/papers/volume22/19-753/19-753.pdf). It permits any comparison arm \(j\); optimality of that arm is not a hypothesis. Its hypotheses include importance-weighted estimators and \(\eta_t\xi_i\le1/4\) for every arm.
With symmetric regularization, \(\xi_i=1\), and Tsallis parameter \(1/2\), the paper’s scaled regularizer agrees with \(F_t\) up to a constant on the simplex when \(\eta_t=1/(4\sqrt t)\). Such a constant cancels from the within-round potential difference. The learning-rate condition holds even at \(t=1\).
Substitution gives equation (2), including its coefficient \(3\). Using \(u_{t,b}\le\sqrt{u_{t,b}}\) gives the coefficient \(1/2\) in equation (3). The cited theorem is correctly applied.
The importance-weighted estimates are unbiased in the stated bandit setting. Since \(F_T(e_j)=0\), the terminal potential bound has the correct direction.
The telescoping calculation correctly retains the changes between consecutive regularizers. Evaluating the previous potential at the current optimizer gives the stated upper bound on each such change.
The inequalities involving \(\sum_b\sqrt{w_{t,b}}-1\), Jensen’s inequality, and \(\sqrt t-\sqrt{t-1}\) are valid. Combining the penalty coefficient \(16\) with stability coefficient \(1/2\), then enlarging to \(17\), proves equation (1).
Cauchy–Schwarz over the rounds and the \(m-1\) noncomparison arms gives equation (4), with precisely the factor \((m-1)H_Tz_j(T)\).
Under the stipulated probability-law evolution, aggregate proposal flows are deterministic. A single nonatomic individual cannot alter those flows.
Every private action probability is positive, so its population average is positive; positive type masses then ensure that each acceptance ratio is defined.
History-independent rationing and fresh reward samples justify
\[
v_{k,b}^t=\nu_{k,b}f_{k,b}^t
\]
conditional on any private history and the selected arm. The potential-loss-vector construction reproduces the observable feedback distribution and supplies the bandit model required by the preceding estimate.
This comparison holds the actual aggregate flows fixed. That is the comparison used throughout the proof; it does not require a counterfactual population-wide deviation.
The witness priority condition excludes all later types from having priority above \(a_k\) at \(b_k\). Every earlier type’s proposal mass to \(b_k\) is bounded by its incorrect proposal mass. Hence the higher-priority mass is bounded by \(s_{k-1}^t\).
The resulting acceptance lower bound is valid both when that mass is below capacity and when it exceeds capacity. Equation (6) follows using \(\nu_{k,b_k}\le1\).
At an earlier partner \(b_i\), type \(a_i\) outranks \(a_k\), and its intended proposal mass does not exceed capacity. Therefore the accepted-mass bound \(q_{a_kb_i}^t\le e_i^t\) is valid and proves equation (7).
At later partners, the witness preference condition and distinct means give transformed mean gaps of at least \(g=\Delta/2\). Earlier-partner coefficients are at least \(1/2\ge g\), since rewards in \([0,1]\) imply \(\Delta\le1\).
These statements establish equation (5). The proof correctly bounds accepted mass at earlier partners; it does not assume a pointwise acceptance-probability gap there.
The identity \(z_k(T)=E_k(T)/\rho_k\) and the summed form of equation (5) are correct.
Young’s inequality gives the stated coefficient \(289(m-1)H_T/(2g)\). Rearrangement yields equation (8), including the disturbance multiplier \(4/g\). Multiplication by \(\rho_k\) correctly cancels the inverse mass.
The recursion for \(P_k(T)\), its unrolling, and the use of \(\sum_k\rho_k=1\) establish equation (9).
The capacity bound on rejected correct proposals remains valid when its right-hand side is negative. Summing it counts \(E_i(T)\) exactly \(K-i+1\) times, proving equation (10).
For integer \(T\ge1\), the harmonic-number and logarithm bounds used in the final calculation hold. Substitution of \(g=\Delta/2\) gives coefficients \(2312\) and \(128\), and their enlargement to \(2500m/\Delta^2\) is valid. Thus the announced coefficient
\[
\frac{2500K^2}{\Delta^2}
\left(1+\frac8\Delta\right)^{K-1}
\]
is justified and independent of other instance details, including \(\rho_{\min}\).
The scalar normalization formula and its bracket agree with the policy’s first-order conditions. The function is strictly decreasing, so the normalization solution is unique. The stated bisection cost is consistent with that bracket and the sensitivity of the probabilities.
The write-up distinguishes numerical evaluation from the exact policy and explicitly conditions exact-real costs on normalization cost \(Q\). It also acknowledges the missing reward representation needed for bit-operation costs. These qualifications prevent an unsupported finite-precision or bit-complexity claim.
No additional mathematical clarification or repair is required. The complexity-status field below is unresolved only because the output does not establish a standard complexity-class classification; the policy-existence question is settled.
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Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.