Answer: \(\mathrm{NP}\)-hard, even when \(\tau=|\mathcal T|=b=d=1\), while satisfying the promise \(w_1>0\).
Theorem 4 — class: algorithmic
PDF text-layer excerpt — equations and symbol order may be approximateConsider training data $\{ (\theta^1, o_1^1), \ldots, (\theta^\ell, o_1^\ell) \}$. Let $g$ be an outcome function such that $g_1(\theta^k) = o_1^k$ for all $k$. Let $w, \xi^k$ be the weight vector and slack variables output by Training Problem 2, with $w_1 > 0$. Consider corresponding mechanism $(g, p_w)$.
statement extracted from the paper’s text layer; consult the paper for the exact wording
Given a finite complete type set \(\mathcal T\), rational population mass \(\mu\), per-capita resource supplies \(\rho\), the score-ordered greedy mass allocation \(q^\mu\), and a specified symmetric tagged-agent residual and tie-breaking kernel \(G^{\mathrm{tag}}_\mu(t,\widehat t)\), compute \(w\) and slacks \(\xi_{t,S}\ge 0\) minimizing \(\frac12\lVert w\rVert_2^2+C\sum_{t,S}q^\mu_{t,S}\xi_{t,S}\), subject to \(f_w(\mu,t,S)-f_w(\mu,t,S')\ge \mathcal L(S,S')-\xi_{t,S}\) for every positive-mass \((t,S)\) and every alternative bundle \(S'\), with \(w_1>0\), and output \(p_w(\mu,S)=-w_{-1}^{\mathsf T}\psi(\mu,S)/w_1\) together with the induced tagged-agent population regret certificate.
Let \(m\geq1\) be the number of items. The finite admissible report domain \(\mathcal T\) is explicitly listed; each type specifies at most \(b\) nonempty desired bundles with nonnegative binary-rational values, and \(v_t(S)\) is the maximum listed value whose bundle is contained in \(S\), or zero if none exists. All bundles \(S\subseteq[m]\), including \(\emptyset\), are possible outcomes.
A society is given by its support: the list of the \(\tau\) types with positive mass, each mass a binary rational. Here \(\tau\) is the support size, not the number of possible types; \(\sum_t\mu_t=1\). Types outside the support may still be admissible reports.
Supplies \(\rho_j\geq0\) are binary-rational quantities per unit population. Greedy mass allocation processes desired type–bundle pairs in descending order of \(v_t(S)/\sqrt{|S|}\), with an explicitly supplied total order resolving ties; it assigns the largest mass allowed by the type's remaining unassigned mass and the remaining supply of every item in the bundle. Unassigned mass receives \(\emptyset\). The resulting \(q^\mu\) is supplied as a sparse rational table, with \(\sum_Sq^\mu_{t,S}=\mu_t\).
For every supported true type \(t\) and admissible report \(\widehat t\in\mathcal T\), an explicitly supplied sparse rational probability distribution \(G^{\mathrm{tag}}_\mu(t,\widehat t;S)\) specifies the tagged agent's bundle under the fixed residual population and tie-breaking convention. It depends on types, not agent identities, and satisfies \(G^{\mathrm{tag}}_\mu(t,t;S)=q^\mu_{t,S}/\mu_t\). A single tag's report leaves \(\mu\) and the price context unchanged.
A supplied Boolean circuit evaluates \(\psi(\mu,S)\in\mathbb Q^d\), with each coordinate encoded using at most \(L\) bits; features are independent of the tag's report. Set \(f_w(\mu,t,S)=w_1v_t(S)+w_{-1}^{\mathsf T}\psi(\mu,S)\), and use loss \(\mathcal L(S,S')=\mathbf1_{S\ne S'}\) and binary-rational \(C>0\).
Minimize \(\frac12\lVert w\rVert_2^2+C\sum_{t,S:q^\mu_{t,S}>0}q^\mu_{t,S}\xi_{t,S}\) over real weights and nonnegative slacks, subject to \(f_w(\mu,t,S)-f_w(\mu,t,S')\geq\mathcal L(S,S')-\xi_{t,S}\) for every positive-mass pair and every \(S'\subseteq[m]\). Promise that the minimizer without a sign restriction has \(w_1>0\); the requested optimizer must satisfy this condition.
Output the optimizer, the succinct signed price function \(p_w(\mu,S)=-w_{-1}^{\mathsf T}\psi(\mu,S)/w_1\), and the population regret certificate \(B=\sum_{t,S}q^\mu_{t,S}\xi_{t,S}/w_1\). Prices are used without clipping or normalization. Tagged regret means \(R=\sum_t\mu_t[\max_{\widehat t\in\mathcal T}\sum_SG^{\mathrm{tag}}_\mu(t,\widehat t;S)(v_t(S)-p_w(\mu,S))-\sum_SG^{\mathrm{tag}}_\mu(t,t;S)(v_t(S)-p_w(\mu,S))]\); the certificate states \(R\leq B\).
Is this exact population training and payment-certificate problem solvable in polynomial time, or is it NP-hard? State running times in \(m,\tau,L,|\mathcal T|,b,d\), where \(L\) is the total input bit length.
For finite-agent training, feasible slacks certify training-example regret at most \(\xi/w_1\) when \(w_1>0\); this is a conditional regret bound, not a polynomial-time training theorem. The full version retains this condition and discards nonpositive learned coefficients. [Source](https://webspace.maths.qmul.ac.uk/felix.fischer/publications/dfjo_svmmd.pdf).
No result settling this precisely specified population problem was located.
The specified problem is \(\mathrm{NP}\)-hard, even when \(\tau=|\mathcal T|=b=d=1\), all masses and supplies equal \(1\), and the feature takes values only in \(\{0,1,2\}\). The reduction satisfies the promise that the unrestricted optimizer has \(w_1>0\).
The population certificate \(R\leq B\) is valid. An exact algorithm obtained by enumerating bundles runs in \(2^{O(m)}\operatorname{poly}(m,\tau,L,|\mathcal T|,b,d)\) bit operations. A polynomial-time algorithm for the general problem would imply \(\mathrm P=\mathrm{NP}\).
Fix any feasible weights and slacks with \(w_1>0\). Define the true type’s utility by
\(u_t(S)=v_t(S)-p_w(\mu,S)=f_w(\mu,t,S)/w_1.\)
This identity uses the specified signed prices directly.
For every positive-mass training pair \((t,S)\) and every bundle \(S'\), feasibility implies
\(w_1\bigl(u_t(S')-u_t(S)\bigr)\leq \xi_{t,S}-\mathbf 1_{S\ne S'}\leq \xi_{t,S}.\)
Fix a supported type \(t\) and an admissible report \(\widehat t\). Average this inequality over \(S'\) using \(G^{\mathrm{tag}}_\mu(t,\widehat t;S')\), and then over \(S\) using \(q^\mu_{t,S}/\mu_t\). Both are probability distributions. We obtain
\(\displaystyle \sum_{S'}G^{\mathrm{tag}}_\mu(t,\widehat t;S')u_t(S') -\sum_S\frac{q^\mu_{t,S}}{\mu_t}u_t(S) \leq \frac{1}{\mu_t w_1}\sum_Sq^\mu_{t,S}\xi_{t,S}. \)
By the promised truthful-tag identity, the second term is precisely truthful expected utility. The right-hand side does not depend on \(\widehat t\), so the inequality remains valid after maximizing over admissible reports. Multiplying by \(\mu_t\) and summing over supported types proves
\(\displaystyle R\leq \frac{1}{w_1}\sum_{t,S}q^\mu_{t,S}\xi_{t,S}=B.\)
Also \(R\geq0\), because truthful reporting is admissible. This proof requires neither deterministic tagged allocations nor restrictions on the signs of prices.
Let \(\varphi\) be a Boolean formula on \(n\geq1\) variables. We construct a training instance in polynomial time.
Set \(m=n+1\). Let item \(n+1\) be a marker, and write \(A=[m]\).
There is one admissible type \(t\), with mass \(\mu_t=1\). Its only desired bundle is \(A\), valued at \(1\). Thus
\(v_t(A)=1,\qquad v_t(S)=0\quad\text{for every }S\ne A.\)
Give every item supply \(1\). The greedy procedure has only one desired type–bundle pair to process, so it allocates all mass to \(A\). Supply the sparse table
\(q^\mu_{t,A}=1\)
and the tagged distribution
\(G^{\mathrm{tag}}_\mu(t,t;A)=1.\)
These tables satisfy every allocation and truthful-tag requirement. Set \(C=2\).
Use one feature coordinate. For a bundle \(S\), let \(x_j=\mathbf1_{j\in S}\), for \(j\in[n]\). Define the circuit by the following ordered cases:
The circuit ignores \(\mu\), is independent of the tag’s report, and has polynomial size. Every output is a constant-size binary rational.
Two properties will be used:
The second property holds because every Boolean assignment is represented by a bundle contained in \([n]\), which necessarily omits the marker.
Write \(a=w_1\), \(z=w_{-1}\), and \(\xi=\xi_{t,A}\). There is one training example, of mass \(1\), so the objective is
\(\displaystyle F(a,z,\xi)=\frac12(a^2+z^2)+2\xi.\)
For every proper bundle \(S'\), its valuation is zero and its loss against \(A\) is one. Consequently its constraint is
\(a+z\bigl(1-\psi(\mu,S')\bigr)\geq1-\xi.\)
The constraint for \(S'=A\) merely repeats \(\xi\geq0\).
If \(\varphi\) is unsatisfiable, every proper bundle has feature \(0\). The complete feasible system is therefore
\(a+z\geq1-\xi,\qquad \xi\geq0.\)
For \(0\leq\xi\leq1\), Cauchy–Schwarz gives
\(\displaystyle a^2+z^2\geq\frac{(a+z)^2}{2}\geq\frac{(1-\xi)^2}{2}.\)
Hence
\(\displaystyle F(a,z,\xi)\geq\frac{(1-\xi)^2}{4}+2\xi =\frac14+\frac32\xi+\frac14\xi^2. \)
This expression has its unique minimum on that interval at \(\xi=0\). For \(\xi\geq1\), the objective is at least \(2\), so no such point is optimal.
At \(\xi=0\), equality in the lower bound requires \(a=z=1/2\), and these weights are feasible. Thus the unique optimizer is
\(\displaystyle (a,z,\xi)=\left(\frac12,\frac12,0\right),\qquad F^*=\frac14.\)
In particular, \(w_1=1/2>0\).
If \(\varphi\) is satisfiable, a proper bundle with feature \(2\) exists. A proper bundle with feature \(0\) also exists, namely \(\{n+1\}\). Thus the complete feasible system is
\(a+z\geq1-\xi,\qquad a-z\geq1-\xi,\qquad \xi\geq0.\)
For \(0\leq\xi\leq1\), the first two inequalities imply
\(a\geq1-\xi+|z|\geq0.\)
Therefore \(a^2+z^2\geq(1-\xi)^2\), with equality only when \(a=1-\xi\) and \(z=0\). It follows that
\(\displaystyle F(a,z,\xi)\geq\frac12(1-\xi)^2+2\xi =\frac12+\xi+\frac12\xi^2. \)
This expression has its unique minimum on the interval at \(\xi=0\). Again, \(\xi\geq1\) gives objective at least \(2\). The unique optimizer is consequently
\((a,z,\xi)=(1,0,0),\qquad F^*=\frac12.\)
Here \(w_1=1>0\).
The construction has polynomial input length and satisfies
\(\displaystyle \varphi\text{ is satisfiable} \quad\Longleftrightarrow\quad w_1^*>\frac34. \)
Both cases satisfy the positivity promise. An exact optimizer therefore decides Boolean satisfiability in polynomial additional time. This establishes \(\mathrm{NP}\)-hardness of the requested output problem.
The learned prices also distinguish the cases: \(p_w(\mu,A)=-1\) in the unsatisfiable case and \(p_w(\mu,A)=0\) in the satisfiable case.
In this reduction, \(B=R=0\), since the slack is zero and the report domain is a singleton. Thus hardness persists even when evaluating the regret certificate itself is trivial.
Let
\(N=\bigl|\{(t,S):q^\mu_{t,S}>0\}\bigr|.\)
The greedy procedure allocates each supported type only its listed desired bundles and possibly \(\emptyset\), so
\(N\leq\tau(b+1).\)
The point \(w=0\), with every slack equal to \(1\), is feasible and has objective \(C\), because the training masses sum to \(1\).
On the feasible objective sublevel set with objective at most \(C\),
\(\|w\|_2^2\leq2C,\qquad 0\leq\xi_{t,S}\leq1/q^\mu_{t,S}.\)
This set is nonempty, closed, and bounded in a finite-dimensional space. It is therefore compact, and the continuous objective attains its minimum.
Any two optimizers with different weights would have a feasible midpoint with strictly smaller objective, by strict convexity of the squared norm and linearity of the slack penalty. Thus the optimal weights are unique.
For fixed weights, every slack has a positive objective coefficient and must equal its smallest feasible value:
\(\displaystyle \xi_{t,S} = \max\left\{0,\max_{S'\subseteq[m]} \left[\mathbf1_{S\ne S'}+f_w(\mu,t,S')-f_w(\mu,t,S)\right]\right\}. \)
Hence the slacks, and therefore the entire optimizer, are unique.
The positivity promise means this unrestricted optimizer is also the requested optimizer satisfying \(w_1>0\).
For completeness, the exponentially many constraints do not force an exponentially long numerical output.
Write the expanded program as minimizing
\(F(x)=\frac12x^{\mathsf T}Qx+c^{\mathsf T}x\)
subject to \(Ax\geq h\), where \(x=(w,\xi)\) has dimension \(k=d+1+N\), and \(Q=\operatorname{diag}(I_{d+1},0_N)\). Every coefficient has bit length \(O(L)\).
At the unique optimizer \(x^*\), choose a maximal linearly independent collection \(E\) of active constraint rows, with right-hand side \(h_E\). It has at most \(k\) rows.
For every \(u\in\ker E\), sufficiently small movements \(x^*\pm\varepsilon u\) are feasible: all active rows remain equalities, and the finitely many inactive inequalities retain their positive gaps. Optimality therefore implies
\((Qx^*+c)^{\mathsf T}u=0.\)
Thus some vector \(\lambda\) satisfies
\(Qx^*+c=E^{\mathsf T}\lambda,\qquad Ex^*=h_E.\)
The coefficient matrix of this linear system is nonsingular. Indeed, a homogeneous solution \((u,v)\) satisfies \(Qu=E^{\mathsf T}v\) and \(Eu=0\). Multiplication by \(u^{\mathsf T}\) gives \(u^{\mathsf T}Qu=0\), hence \(Qu=0\). Independence of the rows of \(E\) then gives \(v=0\). If \(u\ne0\), small feasible movements along \(u\) would leave the objective unchanged, contradicting uniqueness. Therefore \(u=0\).
This nonsingular rational system has dimension at most \(2k\). Clearing denominators and applying determinant bounds shows that its solution has bit length \(\operatorname{poly}(k,L)\). Consequently the optimizer, its price coefficients, and \(B\) all have polynomial-length rational representations in the original parameters.
Enumerating all bundles produces a rational convex quadratic program with
\(k=d+1+N\)
variables and at most
\(M=N2^m+N\)
inequalities, including nonnegativity of the slacks.
Evaluating a valuation requires at most \(b\) bundle-containment checks. Evaluating the supplied feature circuit takes polynomial time in \(L\). Thus constructing the expanded program takes \(2^m\operatorname{poly}(m,\tau,b,d,L)\) time, and its encoding length is \(2^m\operatorname{poly}(m,\tau,b,d,L)\).
The standard polynomial-time theorem for rational convex quadratic programming applies: the objective has a rational positive-semidefinite quadratic matrix, the constraints are rational and linear, and a finite attained optimum exists as proved above. It yields an exact solution in time polynomial in the expanded encoding length. See [Kozlov, Tarasov, and Khachiyan, *The polynomial solvability of convex quadratic programming*](https://www.mathnet.ru/eng/zvmmf5189).
The resulting bound, including all supplied input, is
\(\displaystyle 2^{O(m)}\operatorname{poly}(m,\tau,L,|\mathcal T|,b,d).\)
In particular, training is polynomial-time solvable for fixed \(m\).
Once the weights are known:
These computations take \(\operatorname{poly}(m,\tau,L,|\mathcal T|,b,d)\) time. The total input length \(L\) includes every tagged-distribution entry, including those for reports outside the population support.
The cited paper’s Theorem 4.1 gives the finite-agent slack bound conditional on \(w_1>0\); its subsequent discussion discards learned hypotheses with nonpositive \(w_1\). Its tractable training result imposes additional feature and weight structure. These statements are consistent with the hardness above for arbitrary supplied feature circuits. See [Dütting et al., *Payment Rules through Discriminant-Based Classifiers*, Theorems 4.1 and 5.2](https://webspace.maths.qmul.ac.uk/felix.fischer/publications/dfjo_svmmd.pdf).
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The prover’s claim is proved. The reduction establishes NP-hardness of the exact optimizer output problem while satisfying the positivity promise. The population regret certificate, rational-output argument, and stated exponential-time upper bound also check out. No major or minor mathematical issues were found.
The identity \(u_t(S)=f_w(\mu,t,S)/w_1\) follows directly from the specified signed price function. Dividing the feasibility inequalities by \(w_1\) is legitimate because this section explicitly assumes \(w_1>0\).
The averaging step uses inequalities valid for every competing bundle \(S'\), and only averages training bundles \(S\) having positive mass. For a supported type, \(\mu_t>0\), so \(q^\mu_{t,S}/\mu_t\) is well-defined and sums to one.
The supplied tagged distribution is also a probability distribution. Averaging against these two distributions requires no coupling or independence assumption about actual allocations; it is simply a weighted sum of pointwise inequalities.
The truthful-tag identity correctly identifies the truthful expected utility. The bound is uniform over admissible reports, so maximizing over reports preserves it. Multiplication by \(\mu_t\) and summation give the stated \(R\leq B\).
Truthful reporting belongs to the admissible report domain, which also justifies \(R\geq0\). Reports outside the population support are covered by the argument.
The construction has \(m=n+1\geq2\), one supported type, one admissible report, one desired bundle, and one feature coordinate. The desired bundle \(A=[m]\) is nonempty, and its value is a permitted nonnegative binary rational.
Because a bundle contains \(A\) precisely when it equals \(A\), the asserted valuation is correct: \(v_t(A)=1\), and every proper bundle has value zero.
There is only one desired type–bundle pair. With type mass and all item supplies equal to one, greedy allocation assigns its entire mass to \(A\). The supplied allocation and tagged tables therefore satisfy the required normalization and truthful-tag identity. The singleton processing order also removes any tie-breaking ambiguity.
The construction uses no inadmissible restriction on possible outcomes: every bundle, including \(\emptyset\), remains available in the training constraints.
The ordered cases ensure that \(\psi(\mu,A)=1\), regardless of the formula’s value on the all-ones assignment. Every proper bundle has feature zero or two.
The bundle \(\{n+1\}\) is proper because \(n\geq1\). It contains the marker and therefore has feature zero.
Every Boolean assignment corresponds to a bundle contained in \([n]\), including the all-zero assignment represented by \(\emptyset\). Such bundles omit the marker and are necessarily proper. Consequently, a proper bundle with feature two exists exactly when the formula is satisfiable.
The circuit can implement these tests and evaluate the input formula with polynomial size. It ignores the population input and remains independent of the tagged agent’s report.
The single positive-mass training pair is \((t,A)\), with mass one and \(C=2\). Hence the objective is exactly \(F(a,z,\xi)=\tfrac12(a^2+z^2)+2\xi\).
For a proper competing bundle, the valuation difference is one, the feature difference is \(1-\psi(\mu,S')\), and the loss is one. Thus the stated constraint \(a+z(1-\psi(\mu,S'))\geq1-\xi\) has the correct signs and constant.
The competing bundle \(A\) contributes only \(0\geq-\xi\), which is equivalent to slack nonnegativity. No constraint category is omitted.
Unsatisfiability makes every proper-bundle feature zero, so the full system reduces to \(a+z\geq1-\xi\) and \(\xi\geq0\).
On \(0\leq\xi\leq1\), the quantity \(1-\xi\) is nonnegative. Therefore the proof’s squaring step is valid: \((a+z)^2\geq(1-\xi)^2\). Cauchy–Schwarz gives the other inequality used in the lower bound.
The expansion \(\tfrac14(1-\xi)^2+2\xi=\tfrac14+\tfrac32\xi+\tfrac14\xi^2\) is correct and is uniquely minimized at \(\xi=0\) on that interval. For \(\xi\geq1\), the penalty alone is at least two, exceeding the exhibited feasible objective value.
The equality conditions correctly force \(a=z=\tfrac12\). Thus the unique optimizer and objective value are exactly those claimed, and \(w_1>0\).
Both feature values zero and two occur on proper bundles, producing exactly the two stated inequalities. Their conjunction is equivalent to \(a\geq1-\xi+|z|\).
On \(0\leq\xi\leq1\), this implies the claimed norm lower bound. Its equality conditions are correctly stated. The expansion \(\tfrac12(1-\xi)^2+2\xi=\tfrac12+\xi+\tfrac12\xi^2\) is correct and uniquely minimized at zero slack. The exclusion of \(\xi\geq1\) is again valid.
Consequently, the unique optimizer is \((1,0,0)\). Both satisfiability cases satisfy the unrestricted-optimizer positivity promise.
The threshold test \(w_1^*>\tfrac34\) distinguishes the two cases exactly. The instance construction is polynomial, and inspecting the requested exact optimizer requires polynomial additional time. This establishes the claimed NP-hardness of the output problem.
The price values at \(A\) also check out. Finally, \(B=0\) follows from zero slack, while \(R=0\) follows from the singleton report domain. These observations are consistent with the reduction and do not substitute a degenerate regret problem for the requested training problem.
The allocation procedure supports at most the listed desired bundles and possibly \(\emptyset\) for each supported type, giving \(N\leq\tau(b+1)\).
The proposed feasible point has objective \(C\), since all training masses sum to one. On the corresponding objective sublevel set, the stated weight and slack bounds follow from nonnegativity of every objective term and positivity of every included training mass.
The feasible sublevel set is closed and bounded in finite dimension, so the compactness argument establishes attainment.
Strict convexity in the weights excludes distinct optimal weight vectors. For fixed weights, every slack has a strictly positive penalty coefficient and therefore must equal its smallest feasible value. The displayed maximum gives precisely that value. Together these facts establish uniqueness of the whole optimizer.
The positivity promise then makes the unrestricted optimizer feasible for the requested strict-sign condition.
The expanded constraints and objective have rational coefficients of the stated bit-length order. The number of variables is \(k=d+1+N\).
A maximal independent selection of active rows spans all active rows. Thus directions in \(\ker E\) preserve all active equalities. Because the expanded constraint family is finite, sufficiently small movements in either direction also preserve every inactive inequality. The stationarity argument is therefore valid.
The nonsingularity argument correctly uses positive semidefiniteness of \(Q\), independence of the rows of \(E\), and uniqueness of the optimizer. The stationarity relation also ensures that the proposed flat feasible directions would leave the linear objective term unchanged.
The resulting rational linear system has dimension at most \(2k\). Standard denominator clearing and determinant bounds give polynomial coordinate bit length, independent of the number of expanded constraints. Division by the positive rational \(w_1\) and the finite weighted sum defining \(B\) preserve polynomial representation length.
The variable count and inequality count correctly include all competing bundles and slack nonnegativity. Valuation evaluation and circuit evaluation have the claimed polynomial costs.
The invoked exact convex quadratic-programming result applies to rational linear constraints and a rational positive-semidefinite quadratic objective with a finite attained optimum. Those hypotheses are established in the write-up. The cited result concerns exact polynomial-time computation in binary input length. [Kozlov, Tarasov, and Khachiyan](https://www.mathnet.ru/php/archive.phtml?jrnid=zvmmf&option_lang=rus&paperid=5189&wshow=paper).
Applying this theorem to an expanded encoding of length \(2^m\operatorname{poly}(m,\tau,b,d,L)\) gives the stated \(2^{O(m)}\operatorname{poly}(m,\tau,L,|\mathcal T|,b,d)\) bound. Polynomial time for fixed \(m\) is consistent with general NP-hardness.
The post-training computations use only polynomial-length rational coefficients and explicitly supplied supports. Their claimed polynomial bounds correctly count the tagged-distribution tables as part of \(L\).
The description of the finite-agent source is accurate: Theorem 4.1 conditions its regret bound on \(w_1>0\), the subsequent discussion discards nonpositive learned coefficients, and Theorem 5.2 imposes additional valuation, feature, and weight structure. None establishes tractability for arbitrary supplied feature circuits. [Dütting et al., Theorems 4.1 and 5.2](https://webspace.maths.qmul.ac.uk/felix.fischer/publications/dfjo_svmmd.pdf).
Actionable feedback: No mathematical repair or additional clarification is required.
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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.