Problems

Fix a positive exponent parameter smaller than one hundredth. Consider the fixed smoothly forced whole-space incompressible Navier–Stokes singular solution described in the cited September 2026 source, normalized so that its singular time is one. Measure time by the remaining time before the singularity. The singular core has radial size proportional to the square root of the remaining time, axial size proportional to the remaining time raised to one half minus the exponent parameter, dominant azimuthal and axial speeds proportional to the remaining time raised to minus one half minus that parameter, and radial speed no larger in order than the inverse square root of the remaining time. For each sufficiently small positive low-Mach parameter, consider the standard barotropic compressible Navier–Stokes equations in three spatial dimensions with fixed positive shear viscosity, bulk viscosity satisfying the usual nonnegative dissipation condition, and a smooth pressure law whose derivative equals one at unit density and stays positive near unit density. Scale the pressure force by the inverse square of the low-Mach parameter. Use exactly the same smooth compactly supported external force as in the fixed incompressible singular solution, without retuning that force as the low-Mach parameter varies. Start from zero velocity and from a density equal to one plus a perturbation of quadratic size in the low-Mach parameter, with the perturbations uniformly bounded in one fixed Sobolev space of order at least six and with the density uniformly positive. The critical exponent is defined to be two divided by one plus twice the exponent parameter. For every smaller exponent, determine whether the initial density perturbation can be chosen so that the compressible solution remains smooth until the time whose distance from the singular time is the low-Mach parameter raised to that exponent, and so that the compressible velocity shadows the incompressible singular profile throughout the shrinking anisotropic core. Shadowing means that the radial velocity error, multiplied by the square root of the remaining time, tends uniformly to zero, while the azimuthal and axial velocity errors, each multiplied by the remaining time raised to one half plus the exponent parameter, also tend uniformly to zero on every fixed multiple of the shrinking radial and axial core scales. If precritical shadowing holds for every exponent below the critical exponent, determine whether the critical regime, where the remaining time equals the low-Mach parameter raised to that critical exponent, has a nontrivial compressible transition limit after rescaling space by the radial and axial core lengths. In that limit, require the low-Mach parameter times the velocity magnitude to remain of order one. Determine whether the limiting compressible dynamics remains smooth or develops a different singularity. If shadowing fails before the critical regime, identify the earliest failure exponent, or the sharpest rigorous bound on it, and provide a mathematical mechanism for the failure inside this fixed barotropic model. Mere deterioration of estimates known only on fixed time intervals does not count as a negative resolution. The parameter range used here is \((0,1/100)\).

Working over ZF set theory, for a fixed field $K$, does the assertion that every vector space over $K$ has a basis imply the full Axiom of Choice? Determine for which fixed fields $K$ this implication holds. A basis is a linearly independent spanning subset, so every vector is a finite linear combination of basis elements.

For subsets $A,B\subseteq\mathbb N$, let $A\le_{SJT}B$ mean that for every computable nondecreasing unbounded function $h:\mathbb N\to\mathbb N\setminus\{0\}$ there is a uniformly $B$-computably enumerable sequence of finite sets $(T_n)$ with $|T_n|\le h(n)$ such that $J^A(n)\downarrow$ implies $J^A(n)\in T_n$, where $J^A$ is a fixed universal partial $A$-computable jump function. Let $\mathrm{MLR}^X$ denote the Martin-Lof random reals relative to $X$. Must $A\le_{SJT}B$ imply $\mathrm{MLR}^B\subseteq\mathrm{MLR}^A$?

Does there exist a partial computable map $f:\subseteq 2^\omega\to 2^\omega$ that is injective on its domain, takes each Martin-Lof random real in its domain to a Martin-Lof random real, and satisfies $\mu(\operatorname{dom}f)>0$ and $\mu(\{x\in\operatorname{dom}f:f(x)<_T x\})>0$? Here $\mu$ is the fair-coin product measure, and $f(x)<_T x$ means that $f(x)$ is Turing reducible to $x$ while $x$ is not Turing reducible to $f(x)$.

Let $M$ be a countable transitive model of ZFC. Let $C_M$ be the set of reals in $2^\omega$ that are Cohen-generic over $M$, with its standard Borel structure. For $x,y\in C_M$, put $x\,E_M\,y$ if and only if $M[x]=M[y]$. Is $E_M$ hyperfinite for every such $M$? Explicitly, must there be Borel equivalence relations $F_0\subseteq F_1\subseteq\cdots$ on $C_M$, each having only finite equivalence classes, such that $E_M=\bigcup_{n\in\mathbb N}F_n$?

Give $S_\infty$, the group of permutations of $\mathbb N$, the pointwise convergence topology. Call a subgroup $G\le S_\infty$ finitely periodic if for every nonidentity $g\in G$ the cyclic group $\langle g\rangle$ has only finitely many finite orbits on $\mathbb N$. Call $G$ maximal finitely periodic if no proper supergroup of $G$ in $S_\infty$ is finitely periodic. Can such a maximal group be countable, a countable union of compact sets, or Borel? Determine existence in each of these three classes.

Does ZFC prove that there exists a set $E\subseteq\mathbb R^2$ of Hausdorff dimension $1$ such that for every line $L$ through the origin the orthogonal projection of $E$ onto $L$ has Hausdorff dimension $0$? No Borel or analytic regularity is required of $E$.

Let $E$ be equality on $\mathbb R$ and let $F$ be the equivalence relation on $\mathbb R^\mathbb N$ defined by equality of the countable sets enumerated: $x F y$ if and only if $\{x_n:n\in\mathbb N\}=\{y_n:n\in\mathbb N\}$. For equivalence relations on standard Borel spaces, $R\le_B S$ means that some Borel map $f$ satisfies $u R v$ if and only if $f(u) S f(v)$; write $R<_B S$ if $R\le_B S$ and $S\not\le_B R$. Does there exist a complete first-order theory $T$ in a countable language such that $E<_B\cong_T<_B F$, where $\cong_T$ is isomorphism on the standard Borel space of models of $T$ with universe $\mathbb N$?

Determine whether a balanced constant schedule is $s$-composable at every horizon.

$\mathbb{G}^{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$.

Let \(p_j\) be the \(j\)-th prime and let \(d_j=p_{j+1}-p_j\). Is it true that every positive integer occurs infinitely often among the values \(\operatorname{lcm}(d_j,d_{j+1})/\gcd(d_j,d_{j+1})\), for \(j\geq1\)?

Let \(p_j\) be the \(j\)-th prime. Is it true that for every integer \(r\geq1\), there is an integer \(j\geq2\) such that \(p_{j+1}-p_j=r(p_j-p_{j-1})\)?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. For a prime $p\equiv1\pmod4$, let $G_p$ be the graph on $\mathbb F_p$ with $u$ adjacent to $v$ exactly when $u-v$ is a nonzero quadratic residue. Let $G_{p,2}$ be the subgraph induced by the common neighbors of $0$ and $1$, and let $\overline{G_{p,2}}$ be its simple graph complement. Is $\vartheta(\overline{G_{p,2}})\le(2/3)\sqrt p$ for every sufficiently large prime $p\equiv1\pmod4$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. For a prime $p\equiv1\pmod4$, let $G_p$ be the graph on $\mathbb F_p$ with $u$ adjacent to $v$ exactly when $u-v$ is a nonzero quadratic residue. Let $G_{p,1}$ be the subgraph induced by the neighbors of $0$, and let $\overline{G_{p,1}}$ be its simple graph complement on that same vertex set. Is $\vartheta(\overline{G_{p,1}})\sim\sqrt{p/2}$ as $p\to\infty$ through primes congruent to $1$ modulo $4$?

For an integer $M>1$, let $A\in\mathbb R^{(2M-1)\times M}$ be a matrix such that every $M$ rows span $\mathbb R^M$. For $S\subseteq\{1,\ldots,2M-1\}$ let $A_S$ be the submatrix with those rows, and define $\omega(A)=\min_{\operatorname{rank}(A_{S^c})<M}\sigma_M(A_S)$, where $\sigma_M$ is the smallest of its $M$ singular values. Do there exist universal constants $C>0$ and $0<\beta<1$ such that $\omega(A)\le C\beta^M\max_k\|A_k\|_2$ for every such $M,A$?

For each integer $M\ge2$, put $N=4M-5$ and let $A\in\mathbb C^{N\times M}$ have independent standard complex Gaussian entries. Let $p_M$ be the probability that $x\mapsto |Ax|^2$ is injective modulo global complex phase: $|Ax|^2=|Ay|^2$ implies $y=\lambda x$ for some $|\lambda|=1$. Does $\lim_{M\to\infty}p_M=0$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. Let $G$ be a random circulant graph on $\mathbb Z/n\mathbb Z$. Independently include each unordered difference class $\{k,-k\}$ with $k\ne0$ with probability $1/2$, and join $u,v$ when $u-v$ lies in an included class. Is $\mathbb E\vartheta(G)=(1+o(1))\sqrt n$ as $n\to\infty$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. Let $G\sim G(n,1/2)$, meaning that each unordered pair of distinct vertices is independently an edge with probability $1/2$. Is $\mathbb E\vartheta(G)=(1+o(1))\sqrt n$ as $n\to\infty$?

Let $F$ be a field whose underlying set is a standard Borel space and whose addition and multiplication maps $F\times F\to F$ are Borel measurable. Must there exist an algebraically closed field $K$ whose underlying set is standard Borel and whose field operations are Borel, together with an injective Borel field homomorphism $F\to K$?

Does there exist a Borel subset $X\subseteq\mathbb R^2$ such that every affine line $L\subseteq\mathbb R^2$ meets $X$ in exactly two points, that is, $|X\cap L|=2$ for every affine line $L$?

Call $A\subseteq\mathbb R^2$ a Kakeya set here if for every real slope $a$ it contains an entire affine line $\{(x,ax+b):x\in\mathbb R\}$ for some $b\in\mathbb R$. Call such a set inclusion-minimal if no proper subset has this property. Does every Borel Kakeya set in this entire-line sense contain an inclusion-minimal Kakeya subset? The minimal subset is not required to be Borel.

Is there an effectively closed ($\Pi^0_1$) class $C\subseteq 2^{\mathbb N}$ such that its computable members form an infinite maximal almost disjoint family among the computable subsets of $\mathbb N$? Each member of that family must be infinite, distinct members have finite intersection, and every infinite computable subset of $\mathbb N$ must have infinite intersection with some computable member of $C$.

Let $M$ be a countably infinite $\omega$-categorical structure in a finite first-order language, and let $G=\operatorname{Aut}(M)$ with its pointwise convergence topology. Let $\operatorname{Aut}(G)$ consist of continuous group automorphisms and let $\operatorname{Inn}(G)$ consist of conjugations by elements of $G$. Must $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$ be finite?

Let \(k>1\), let \(G\) be a group, and let \(a_1H_1,\ldots,a_kH_k\) be pairwise disjoint left cosets of subgroups of finite index in \(G\). Must there be indices \(1\leq i<j\leq k\) such that \(\gcd([G:H_i],[G:H_j])\geq k\)?

Fix a rooted, fully bifurcating binary tree \(T\) with \(N\) labeled leaves and an arbitrary symmetric dissimilarity matrix \(D=(D_{ij})\) with \(D_{ij}\ge 0\) and \(D_{ii}=0\). Let \(\mathcal O(T)\) be the set of leaf orderings obtained from a fixed canonical left-to-right ordering by independently swapping the two child subtrees at every internal node. For \(\pi=(\pi_1,\ldots,\pi_N)\in\mathcal O(T)\), define \(C(\pi)=\sum_{k=1}^{N-1}D_{\pi_k\pi_{k+1}}\). Determine whether there is an exact worst-case \(O(N^2)\)-time algorithm that, for every such \(T\) and \(D\), returns an ordering \(\pi^\star\in\arg\min_{\pi\in\mathcal O(T)} C(\pi)\). In particular, determine whether an exact \(O(N^{2-\varepsilon})\)-time algorithm exists for some fixed \(\varepsilon>0\). A full resolution must cover every binary tree topology and every valid symmetric nonnegative \(D\); partial results must state an explicit structural family or computational assumption. Separately, as a bonus direction, determine whether restricting \(D\) to a metric, or to Euclidean distances among points in fixed dimension \(d\), permits an exact genuinely subquadratic algorithm. This metric/Euclidean direction is distinct from the unrestricted-matrix problem: the standard \(\Omega(N^2)\) unread-entry adversary for arbitrary \(D\) does not automatically apply to these restricted distance classes, so a subquadratic miracle is not ruled out by that argument.

Any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $Θ(n^{-2})$ requires $Ω\!\left(\frac{n}{\log n\,\log\log n}\right)$ membership queries.

This question concerns the classical contribution of one specified replica saddle. The preparation and branch below define that contribution. Whether it is the contribution selected by a full quantum gravitational integration cycle is a separate question. This is an explicit restriction of the original intake to a mathematical sector in which the observable and admissibility tests can be fixed before attempting the thin-wall limit. ## 1. Einstein–scalar completions and matched wall data Fix \(G>0\), \(0<\ell_-<\ell_+\), and \(\sigma>0\). The signs label the two sides; the larger radius is the lower-energy vacuum. For \(0<\epsilon\leq\epsilon_0\), the Lorentzian bulk action is \[ I_\epsilon=\frac1{16\pi G}\int\sqrt{-g}\,R -\int\sqrt{-g}\left[\frac12(\nabla\phi)^2+V_\epsilon(\phi)\right]. \] Use Lorentzian signature \((-++)\). The complete equations, required on every smooth Lorentzian region and complexified for the replica solutions, are \[ R_{\mu\nu}-\tfrac12 Rg_{\mu\nu}=8\pi G T_{\mu\nu},\qquad T_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi -g_{\mu\nu}\left(\tfrac12g^{\rho\sigma} \partial_\rho\phi\partial_\sigma\phi+V_\epsilon(\phi)\right), \qquad \frac1{\sqrt{-g}}\partial_\mu \left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi\right) -V'_\epsilon(\phi)=0. \] Here curvature is the Levi-Civita curvature, with convention \(R_{\mu\nu}=2\ell^{-2}g_{\mu\nu}\) for dS3. Equivalently in three dimensions, \(R_{\mu\nu}=8\pi G(\partial_\mu\phi\partial_\nu\phi+ 2V_\epsilon g_{\mu\nu})\). The induced Cauchy data must obey the normal and mixed projections of the Einstein equation as constraints. “On shell” means these equations hold pointwise off the prescribed replica locus, with the explicit junction/distributional conditions at a limiting wall. There are no higher-curvature or derivative matter couplings. Each \(V_\epsilon\) is real analytic on the real field range and has two labeled strict local minima \(\phi_{\pm,\epsilon}\), with \(V_\epsilon(\phi_{\pm,\epsilon})=(8\pi G\ell_\pm^2)^{-1}\). The positions of the minima and the rest of the potential may vary. In particular, their separation is not assumed to stay bounded below as the wall shrinks at fixed tension. A completion supplies a regular compact rotationally symmetric Euclidean solution \(ds_E^2=d\xi^2+a_\epsilon(\xi)^2d\Omega_2^2\), \(0\leq\xi\leq\xi_{c,\epsilon}\), satisfying \[ \phi''+2(a'/a)\phi'=V'_\epsilon(\phi),\qquad (a')^2=1+8\pi G a^2\left(\tfrac12(\phi')^2-V_\epsilon(\phi)\right). \] Here \(a>0\) in the interior, and at either pole the scalar has an even regular expansion and the scale factor has the odd regular expansion \(a=\xi+O(\xi^3)\), or \(a=(\xi_c-\xi)+O((\xi_c-\xi)^3)\), respectively. The solution need not reach a vacuum minimum at a finite Euclidean pole. Write \(D=(G,\ell_-,\ell_+,\sigma,\mathcal B)\). The geometric datum is the tuple \(\mathcal B=(M_0^L,g_0^L,W_L;M_0^E,g_0^E,W_E;\iota_0,\nu_\pm)\). The exact vacuum-piece, wall and junction records are Schemas 24–25. Here \(M_0^L\) is a specified smooth three-manifold with a continuous piecewise smooth Lorentzian metric \(g_0^L\). Its two components separated by the embedded timelike hypersurface \(W_L\) are isometric, by supplied isometries, to specified open subsets of dS3 of the labeled radii. \(M_0^E\) is a specified compact three-manifold obtained by gluing two round spherical caps of those radii along the embedded two-sphere \(W_E\). The cap metrics and gluing maps are supplied and induce the same metric on \(W_E\). They admit the common angular reflection described below. \(\iota_0\) identifies its fixed two-sphere with a spacelike initial slice of \(M_0^L\), matching the induced metric and zero extrinsic curvature, and identifies the wall intersections on that slice. \(\nu_\pm\) are the chosen one-sided normals. Both glued metrics satisfy their respective Euclidean or Lorentzian pure-tension junction equations with these normals. Thus both limiting geometries, their wall embeddings and gluing maps are input data; neither is to be recovered from an unspecified “common spacetime.” Write \(W=W_L\). Its induced metric agrees on both sides and satisfies \([K_{ab}]-h_{ab}[K]=-8\pi G S_{ab}\), with \(S_{ab}=-\sigma h_{ab}\), using that orientation. Use the angular coordinates \(d\Omega_2^2=d\theta^2+\sin^2\theta\,d\varphi^2\), \(0\leq\theta\leq\pi\), \(\varphi\sim\varphi+2\pi\). The reflection is explicitly \(\rho_E(\xi,\theta,\varphi)=(\xi,\pi-\theta,\varphi)\). Its fixed surface is \(\Sigma=\{\theta=\pi/2\}\), and the preparation half is \(0\leq\theta\leq\pi/2\). This is an angular equator, not the radial midpoint \(\xi_c/2\); no equality of the two radial caps or vacua is imposed. On \(\Sigma\), set \(h=d\xi^2+a_\epsilon(\xi)^2d\varphi^2\), \(\phi_\Sigma=\phi_\epsilon(\xi)\), \(K_{ij}=0\) and \(\pi_\phi=0\). Reflection invariance follows directly from the ansatz, so these zero normal derivatives are explicit boundary conditions. Develop precisely these data by the Lorentzian Einstein–scalar Cauchy equations. Require a smooth Lorentzian development reaching the chosen future conformal boundary. This is the preparation of the background; a coordinate rotation alone is not a substitute for this Cauchy matching condition. Matching is defined by the following explicit comparison maps. Each completion supplies diffeomorphisms \(\Psi_\epsilon^L:M_0^L\to M_\epsilon^L\) and \(\Psi_\epsilon^E:M_0^E\to M_\epsilon^E\), preserving orientation, the marked boundary collar and the initial-slice identification \(\iota_0\). All convergence and distributional tests below refer to pullbacks by these maps to the fixed manifolds in \(\mathcal B\). The fields are continuous in \(\epsilon>0\) in \(C^2\) on real compact sets. There is one connected transition tube \(N_\epsilon(W)\), of normal proper half-width at most \(\epsilon\). Outside every fixed neighborhood of \(W\), the metrics converge in \(C^2_{\rm loc}\) to the pieces specified by \(\mathcal B\), and the scalar stress converges to the corresponding vacuum stress. Furthermore, for every compactly supported smooth test tensor \(f^{\mu\nu}\), \[ \lim_{\epsilon\downarrow0}\int (T_{\mu\nu}^{\epsilon}-T_{\mu\nu}^{\mathrm{vac},\epsilon}) f^{\mu\nu}\,d\mathrm{vol}_{g_\epsilon} =-\sigma\int_W h_{\mu\nu}f^{\mu\nu}\,d\mathrm{vol}_h. \] The vacuum term uses the labeled constant vacuum energy on each side of the fixed wall, and \(h_{\mu\nu}\) is extended tangentially. On \(M_0^E\), require the same off-wall \(C^2_{\rm loc}\) and vacuum-stress convergence, and the same test-tensor distributional identity with \(W_E\), its Riemannian induced metric and volume form. The selected Euclidean transition tube shrinks to this specified \(W_E\). Analytic tails outside the shrinking tube are allowed; their vacuum-subtracted stress tends to zero in total variation on each compact set outside the tube. No holomorphic convergence as \(\epsilon\to0\) is assumed. ## 2. Fixed preparation and replica branch The additional comparison datum is \(P=(M,\Sigma,H,b,q,\gamma,z,A,h_0,K,\mathfrak c)\). The arc \(A\) and conformal metric \(h_0\) are components of \(P\) only. There is no independent interval argument. The accompanying Schemas 1–27 are part of this statement and define its mathematical input-record type. In particular q=Q is a single total map on the full input set of Schema 4, with the exact output tuple of Schema 5. Every completion evaluates that same map. Validity and equality are those in Schemas 16 and 26. The following entries specify the geometric and preparation data. - M and Sigma are the marked preparation and initial-slice presentations of Schemas 2, 25 and 26. H is the complete quotient/cut record of Schema 18, with integral relative homology and cyclic gluing defined in Schemas 17–19. The curve and interval are its actual embedded chains; the homology test is transport of its fixed cut surface and class. A and h_0 have the exact types and collar identifications of Schema 23. - b is the boundary-source record of Schemas 20–21. Its metric and scalar maps, domains, replica gluing and continuation are fixed functions evaluated on the indicated parameters and labeled vacuum values. The unreplicated fields agree with the equatorial preparation. Equatorial Cauchy data come from the completion through Schema 4, with the exact matching in Schema 25. No additional independently adjustable boundary source is admitted. - q=Q provides the explicit gauge and additional local boundary equations, normed perturbation space, trace graph, conical expansion record and residual gauge space in Schemas 5–11. Its coefficients may depend on its typed input; the map itself is fixed throughout a comparison. The seed is the unique solution germ defined by Schema 13. No additional selection predicate or choice by an extremal length, on-shell action ordering or entropy is allowed. - \(\gamma:[0,1]\to\mathbb C^k\) is a fixed piecewise analytic path of boundary and time parameters from that seed to the specified Lorentzian preparation, with the exact domain, breakpoints and nonsingular endpoint convention of Schema 22. Continue the seed germ along this path. No reselection after a crossing or change of action ordering is permitted. - \(z\) is a fixed dimensionless boundary defining function: \(z=0\), \(dz\ne0\) at the future boundary and the rescaled induced metric there is \(h_0\). Its normalization and reference scale are common to every completion. The cutoff boundary is \(z=\delta>0\). - \(K\) is a specified compact oriented real three-manifold used to parametrize the preparation/replica action contour. Its boundary has exactly one external component, mapped to the cutoff boundary. The Euclidean cap and Lorentzian part meet across an interior surface, not an external boundary. All exterior boundaries are smooth away from the prescribed replica endpoints. There are no external joints or corners and hence no corner-action terms in this sector. - \(\mathfrak c\) fixes a relative homotopy class of contour embeddings of \(K\), with the boundary and replica locus marked, and the seed embedding in the real Euclidean section. The same class is transported through the coordinate identifications supplied with the completion and along \(\gamma\). It contains no selection by an action value. Every item of \(P\) is specified by maps, functions and boundary conditions, not by a condition involving the desired entropy. Different choices of \(P\) are different instances of the question. The universal assertion below ranges over all such tuples satisfying the following fixed predicates; a proposed proof cannot choose a smaller admissible class after seeing its answer. For each \(0<\epsilon<\epsilon_C\) and \(0<\delta<\delta_0\), with the family germ and fixed collar domain of Schema 12, require that the selected replica-quotient solution exists for \(|n-1|<r_{\epsilon,\delta}\) with \(r_{\epsilon,\delta}>0\), and along all of \(\gamma\). Away from its fixed curve it solves the complexified Euler–Lagrange equations of the action above. At the fixed curve impose exactly the locally finite regular-power expansion and regular-cone tests in Schemas 9–11, including the opening angle 2pi/n, smooth-cover pullback wherever the branch includes an integer replica, and absence of a scalar defect source. This is the explicit conical sector of the question; other asymptotic classes are modified problems. The equivalent auxiliary tension is (n-1)/(4nG). At n=1 the curve is smooth and extremal, homologous to A in the selected background. Require isolated continuation: the linearized gauge-fixed replica boundary problem has zero kernel after its prescribed gauge modes are removed at every point of the path. The complex data of each completion are a tuple \((X_\epsilon,\{U_j,\chi_j,t_{jk}\},O_\epsilon, g_{\epsilon,n},\phi_{\epsilon,n},\Gamma_{\epsilon,n,\delta})\). Here \(X_\epsilon\) is a complex three-manifold; the open sets \(U_j\) cover the contour, \(\chi_j:U_j\to\mathbb C^3\) are holomorphic coordinate maps and \(t_{jk}=\chi_j\chi_k^{-1}\) are their holomorphic overlap maps satisfying the cocycle identities. \(O_\epsilon\subset \mathbb C\) is an open field-value domain containing the scalar image, and \(V_\epsilon:O_\epsilon\to\mathbb C\) is the chosen single-valued holomorphic extension. Metric and scalar are holomorphic on these charts away from the specified replica locus; the metric determinant is nonzero there. Removable angular and pole singularities are checked in regular charts. The real Euclidean and Lorentzian sections embed into \(X_\epsilon\) and agree with the prepared background at \(n=1\). The map \(\Gamma_{\epsilon,n,\delta}:K\to X_\epsilon\) is a smooth oriented real-three-dimensional contour embedding, with smooth external boundary at \(z=\delta\), in the transported class \(\mathfrak c\). Its pullback metric is nondegenerate away from the replica locus. The maps and fields depend analytically on \(n\) off that locus and continuously along \(\gamma\). Supply the maps, their domains, and their dependence on \(\epsilon,n,\delta\), rather than only a statement that a contour exists. Every compact contour segment is contained in a finite union of these open charts. The defining equations and all integrals are pulled back by \(\Gamma\); contours leave no unspecified boundary or corner contribution. Continue square roots and action phases from the positive Euclidean volume element along \(\gamma\). These are pointwise-in- \(\epsilon,\delta\) conditions; uniform domain size, uniform inverse bounds and a uniform replica radius as \(\epsilon\to0\) are not assumptions. Every input instance must satisfy Schemas 1–27, including the full normed-space and trace records, conical remainder tests and exact linearized kernel test. Unspecified record components fail validity. These definitions impose no condition on the value or convergence of the thin-wall entropy. This defines a single classical branch by analytic continuation. It makes no claim that this branch has a nonzero intersection number with any full gravitational path-integral cycle. A KSW metric test alone would not prove such a claim. Establishing a physical integration cycle, including sums and Stokes jumps of contributing saddles, is a separate extension of this sector. ## 3. Observable, subtraction and order of limits With the holomorphic forms and unit normal of Schema 27, define \[ J_{n,\epsilon,\delta}=n\left\{ \lim_{\eta\downarrow0}\int_{K\setminus N_\eta(L)}\Gamma^* \left[\omega_g\left(-\frac{R(g)}{16\pi G} +\frac12g^{ab}\partial_a\phi\partial_b\phi+V_\epsilon(\phi)\right)\right] -\frac1{8\pi G}\int_{\partial K}\Gamma^*(\omega_h K_{\rm ext})\right\}. \] At the real Euclidean seed these are the positive Euclidean volume forms; the displayed Einstein and external GHY signs are negative and the scalar kinetic/potential signs positive. Continue this entire expression, its curvature, normal and volume-form sheet along gamma as specified in Schema 27. The auxiliary tubular radius eta tends to zero at fixed n, epsilon and delta; the finite punctured bulk limit is required to exist. This is the regular-cover convention: no delta-curvature, auxiliary brane-action term, tip subtraction or artificial inner-boundary GHY term is included. The only external boundary is the smooth cutoff boundary. Preparation seams are interior and there are no external corner terms. These choices define the classical action; the auxiliary tension in Section 2 specifies its conical equations and is not added again to J. Set \(F_{\epsilon,\delta}(n)=J_{n,\epsilon,\delta}-nJ_{1,\epsilon,\delta}\). Fix minimal subtraction in the chosen \(z\) collar as follows. For each \(\epsilon>0\), require a finite divergent expansion \[ F_{\epsilon,\delta}(n) =\sum_{(\alpha,k)\in E_\epsilon} c_{\alpha k,\epsilon}(n)\delta^\alpha(\log\delta)^k +F_\epsilon^{\mathrm{ren}}(n)+R_{\epsilon,\delta}(n), \] where the displayed monomials diverge at \(\delta=0\): \(\alpha<0\), \(k\) a nonnegative integer, or \(\alpha=0,k>0\). Coefficients and the finite term are analytic on a disk about \(n=1\). On some smaller closed disk require \(\sup_n(|R_{\epsilon,\delta}|+|\partial_nR_{\epsilon,\delta}|) \to0\) as \(\delta\downarrow0\). Subtract exactly the displayed divergent terms. The finite coefficient uses \(\log\delta\) in the fixed dimensionless collar, and no finite counterterm is allowed. This is a specified finite-part scheme; its expansion and remainder condition are admissibility requirements at each positive \(\epsilon\). Define \[ \mathscr S_{\epsilon,P} =\left.\partial_n F_\epsilon^{\mathrm{ren}}(n)\right|_{n=1}. \] It is a complex number. Formally, if \(\log Z_n=-J_n/\hbar+O(\hbar^0)\), it is the coefficient \(\lim_{\hbar\to0}\hbar S\). The task fixes that classical coefficient: no determinant, bulk entanglement, loop counterterm or higher-order \(\hbar\) term is included. First extract the classical coefficient at fixed thickness and cutoff, then remove \(\delta\), and only then let \(\epsilon\downarrow0\). No interchange with a quantum limit or uniform loop-error estimate is part of the assertion. Any numerical approximation of this classical quantity must instead bound its own error as thickness shrinks. ## 4. The precise question and resolution conditions Let \(\mathcal C(D,P)\) be exactly the completion families satisfying Sections 1–3 and Schemas 1–27, with Valid(D,P), for all sufficiently small positive \(\epsilon\). Membership does not require a thin-wall entropy limit. Call the data realizable when \(\mathcal C(D,P)\ne\varnothing\). Determine the truth of \[ \forall\,(D,P)\ \text{realizable},\quad \exists\,s(D,P)\in\mathbb C\quad \forall\,C\in\mathcal C(D,P),\quad \lim_{\epsilon\downarrow0}\mathscr S^{C}_{\epsilon,P}=s(D,P). \] A positive resolution proves this assertion on the entire class just defined and constructs at least one realizable instance; conditional results that add complex-domain convergence or uniform saddle bounds are partial results. It must identify the limiting classical prescription or give an equivalent characterization of \(s\) from the common data. Nonexistence of every realizable instance is a separate obstruction to this formulation and must not be reported as a universality theorem. A negative resolution gives explicit fixed data and verifies membership for two completions with unequal finite limits, or for one completion whose complex-valued entropy has no finite limit. Unequal limits require a strictly positive separation bound. Failure of a finite limit can be certified by an unbounded sequence of entropy moduli or two thickness sequences tending to zero whose entropies have a positive separation. All quantities must use the same \(P\), collar and finite-part rule. Degeneration that destroys the replica branch at a positive thickness excludes that family; approaching a degeneration only as \(\epsilon\to0\) is allowed and must be analyzed. This is a formal classical thin-wall question. A result interpreted as a physical approximation at fixed \(\hbar\) additionally needs a justified hierarchy between wall thickness and microscopic cutoffs, control of omitted terms, and a physical saddle-cycle argument. A boundary/bulk pseudoentropy identity additionally needs an independently defined boundary transition operator. Those extensions are not acceptance conditions for this restricted classical problem, and are not established by its initial packet.

Let P be a d-dimensional polytope with n facets, and let G(P) denote its 1-skeleton. Determine whether there exists a universal polynomial p in two variables such that, for every d-dimensional polytope P with n facets, the graph diameter satisfies $\operatorname{diam}(G(P))\le p(n,d)$. Prove such a bound with an explicit universal polynomial, or construct a family of d-dimensional polytopes with n facets whose graph diameters eventually exceed every polynomial in n and d.

Let \(\mathfrak T\) be an explicitly specified hierarchy of downstream statistical task classes indexed by unknown smoothness, effective dimension, and task-space complexity, let \(\mathfrak P\) be a class of pretraining task distributions \(\Pi\), and let \(\mathfrak Q(\Pi)\) be a deployment class \(Q\) constrained by an explicitly defined task-level shift functional \(\Delta(\Pi,Q)\). A pretraining sample contains \(N\) independent task draws and total observation/token budget \(D\), with within-task allocation specified. A learner from a concrete capacity-controlled architecture class of complexity at most \(M\) produces a representation and in-context inference rule, and at deployment receives \(K\) labeled context examples from a new task. For each task \(\tau\), let \(R_\tau(f)=\mathbb E_{(X,Y)\sim P_\tau}[\ell(f(X),Y)]\), \(R_\tau^\star=\inf_{f\in\mathcal F_\tau}R_\tau(f)\), and \(\mathcal E_\tau(f)=R_\tau(f)-R_\tau^\star\). Determine a sharp minimax characterization, up to universal constants or logarithmic factors, of the worst-case downstream excess risk as a function of \(N,D,M,K\), intrinsic task complexity, and \(\Delta(\Pi,Q)\), with a matching information-theoretic lower bound. The theorem must identify the representation complexity required for \(\varepsilon\)-universality, a sharp task-diversity threshold separating memorization from genuine task generalization, the maximal admissible distribution shift for \(\varepsilon\)-universality, and the statistically and computationally optimal allocation of resources among task diversity, observations, capacity, and context. The achieving learner must adapt without being given the unknown smoothness, effective dimension, task complexity, or shift radius. The result must include impossibility boundaries when deployment contains directions statistically invisible under pretraining and must state all identifiability, allocation, boundedness, loss, architecture, and edge-case assumptions needed to make the target checkable rather than a claim over arbitrary foundation-model classes.

Determine whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor.

The $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification.

Let \(\Omega(n)\) denote the total number of prime divisors of \(n\), counted with multiplicity. Determine whether there exists an absolute constant \(C>0\) such that every positive integer solution \((a,b,c,x,y,z)\) of \(a^x+b^y=c^z\), with \(x,y,z>2\), satisfying \(\Omega(abc)/(x+y+z)\le C\), necessarily satisfies \(\gcd(a,b,c)=\min(a,b,c)\).

The set of bounded ratios $\BR(X)$ on a semialgebraic set $X\subset\R^n_{>0}$ is the convex cone of linear forms that are nonnegative on the tropicalization $\trop(X)$.

Let \(\mathcal X=[0,1]^d\), \(d\ge1\). For every smoothness index \(r=p+\alpha>1\), where \(p=\lfloor r\rfloor\ge1\) and \(0<\alpha\le1\), define the normalized nonconvex Hölder class \(\mathcal F_r(L)\) to consist of functions \(f:[0,1]^d\to\mathbb R\) such that \(f\in C^{p,\alpha}\), \(\|f\|_\infty\le1\), \(\|D^j f\|_\infty\le L\) for \(1\le j\le p\), the order-\(p\) derivative has Hölder seminorm \([D^p f]_\alpha\le L\), and \(f\) is nonconvex on \([0,1]^d\). A randomized comparison-only algorithm adaptively chooses query points and receives only the pairwise comparison oracle \(\operatorname{Comp}_f(x,y)=1\) if \(f(x)<f(y)\) and \(0\) otherwise. After at most \(N\) comparison queries it outputs \(\widehat x_N\). Determine the sharp minimax simple-regret rate \(R_N^{\mathrm{cmp}}(\mathcal F_r,L,d)=\inf_{A\in\mathfrak A_{\mathrm{cmp}}}\sup_{f\in\mathcal F_r(L)}\mathbb E_A[f(\widehat x_N)-\min_x f(x)]\) for every \(r>1\), including its dependence on \(N,d,L,r\). Prove matching lower and upper bounds up to explicitly characterized logarithmic factors if necessary. Determine whether a single randomized comparison-only algorithm, not given \(r\), can attain these rates simultaneously over a nontrivial interval of \(r\). Determine the corresponding high-probability evaluation complexity and compare the comparison-only rate with the cardinal value-oracle minimax rate, identifying any asymptotic comparison-information penalty. As a secondary structural question, determine whether restricting the algorithm to population-based comparison-only architectures changes the minimax rate.

Let $\mathcal F_{\mathrm{nl}}$ be a class of twice continuously differentiable, nonconvex equality-constrained problems $\min_{x\in\mathbb R^n} f(x)$ subject to $c(x)=0$, where $c:\mathbb R^n\to\mathbb R^m$ is genuinely nonlinear, its Jacobian has uniformly bounded full row rank on the relevant region, and prescribed Lipschitz/Hölder bounds hold for the relevant derivatives. For $L(x,\lambda)=f(x)+\lambda^\top c(x)$, define an $(\epsilon_g,\epsilon_H,\epsilon_c)$-second-order KKT point by the existence of $\lambda$ such that $\|\nabla_xL(x,\lambda)\|\le\epsilon_g$, $\|c(x)\|\le\epsilon_c$, and $v^\top\nabla^2_{xx}L(x,\lambda)v\ge-\epsilon_H\|v\|^2$ for every $v$ satisfying $\nabla c(x)v=0$. Fix a deterministic second-order oracle model specifying access to $f,c,\nabla f,\nabla c$, and second-order information. Determine the minimax evaluation complexity $\mathcal C^*(\epsilon_g,\epsilon_H,\epsilon_c;n,m)=\inf_{\mathcal A}\sup_{P\in\mathcal F_{\mathrm{nl}}}N_{\mathcal A}(P;\epsilon_g,\epsilon_H,\epsilon_c)$. In particular, determine whether an $\Omega(\epsilon_H^{-3})$ lower bound is unavoidable for genuinely nonlinear equality-constrained problems and matches an $O(\epsilon_H^{-3})$ upper bound, determine the sharp dependence on $\epsilon_g$ and $\epsilon_c$, and decide whether a joint $\Theta(\epsilon_g^{-3/2}+\epsilon_H^{-3}+\epsilon_c^{-1})$ law is valid up to constants or logarithmic factors, or whether a different coupled dependence is necessary. A mere reduction using affine constraints is only a baseline lower bound and does not resolve the intrinsic nonlinear-constraint question.

Let \(\Omega(n)\) denote the total number of prime divisors of \(n\), counted with multiplicity, and let \(\omega(n)\) denote the number of distinct prime divisors of \(n\). For every real number \(e>0\), there exists a constant \(C_e\ge1\), depending only on \(e\), such that every positive integer solution \((a,b,c,x,y,z)\) of \(a^x+b^y=c^z\), with \(x,y,z\ge3\), satisfying \(\Omega(abc)\le e(x+y+z)\), satisfies \(\omega(\gcd(a,b,c))\ge \omega(abc)/C_e\).

Given a rational linear program $\min\{c^\top x:Ax=b,\ x\ge0\}$, with $A\in\mathbb Q^{m\times n}$, and a feasible starting point $x^{(0)}$, determine whether there exists a deterministic algorithm that, at every nonoptimal feasible iterate $x^{(k)}$, computes in time polynomial in $m,n$ an improving circuit $g^{(k)}\in\ker A$, chooses the maximal feasible step $\alpha_k=\max\{\alpha\ge0:x^{(k)}+\alpha g^{(k)}\ge0\}$, and sets $x^{(k+1)}=x^{(k)}+\alpha_k g^{(k)}$. The algorithm must reach an optimal solution after at most $\operatorname{poly}(m,n)$ augmentations, with the augmentation bound independent of the numerical magnitudes and encoding length of $A,b,c$, and with polynomially bounded arithmetic complexity of the intermediate iterates. The strongest target is an $O(m^2\log m)$ augmentation bound.

For all $\ell>30$ $$ \ell+1=\s(n,P_\ell)< \s(n,S_\ell)=\binom{\ell}{2}-1$$ where $P_\ell$ and $S_\ell$ denote the path and star on $\ell$ vertices, respectively.

Let \(p_1=2<p_2<p_3<\cdots\) be the sequence of prime numbers in increasing order. For every integer \(m\ge1\), define \(q=p_{m+1}\) and \(P=\prod_{i=1}^{m}p_i\). For a positive integer \(n\) and a prime \(p\), let \(v_p(n)\) denote the largest integer \(k\ge0\) such that \(p^k\mid n\). Define \(T_m:\mathbb N_+\to\mathbb N_+\) by \(T_m(a)=\frac{qa+1}{\prod_{i=1}^{m}p_i^{v_{p_i}(qa+1)}}\). Equivalently, \(T_m(a)\) is obtained from \(p_{m+1}a+1\) by removing every prime-power factor whose prime belongs to \(\{p_1,\ldots,p_m\}\). For \(a_1\in\mathbb N_+\), consider the forward orbit \(a_{n+1}=T_m(a_n)\). Vô Conjecture 7 (VC7): For every \(m\ge1\): (1) every forward orbit of \(T_m\) is eventually periodic; that is, for every \(a_1\in\mathbb N_+\), there exist \(N,T\ge1\) such that \(a_{n+T}=a_n\) for all \(n\ge N\); and (2) for every periodic cycle \(C\) of \(T_m\), one has \(\min C<P=\prod_{i=1}^{m}p_i\).

Determine the asymptotically optimal worst-case straight-line complexity of strictly feasible, bounded linear programs in standard form. For each \(n\), let \(S(n)\) be the supremum, over all \(n\)-variable LPs with full-row-rank constraint matrix and nonempty primal and dual strict interiors, of the minimum number of affine segments in any piecewise-linear curve that remains in the fixed wide neighborhood \(N_{-\infty}(1/2)\) and satisfies \(\mu(\Gamma(\mu))=\mu\) on each finite interval \([\mu_1,\mu_0]\), with \(\operatorname{SLC}_{1/2}(P)=\sup_{0<\mu_1<\mu_0<\infty}\operatorname{SLC}_{1/2}(P;\mu_1,\mu_0)\). Here \(N_{-\infty}(1/2)=\{(x,s)\in P_{++}\times D_{++}:x_i s_i\geq\tfrac12\mu(x,s)\ \forall i\}\), where \(\mu(x,s)=x^\top s/n\). Determine the asymptotic growth of \(S(n)\), in particular whether \(S(n)=2^{\Theta(n)}\), and more broadly whether this straight-line complexity is polynomially equivalent to the intrinsic iteration complexity of an explicitly defined class of wide-neighborhood path-following interior-point methods.

For a bounded, nondegenerate linear program \(\min\{c^\top x:Ax=b,\ x\ge0\}\), with \(A\in\mathbb R^{m\times n}\), does there exist a deterministic pivot rule \(\mathcal R\), computable in time polynomial in the encoded rational input and the current basis, such that from every feasible starting basis the resulting simplex trajectory reaches an optimal basis after at most \((m+n)^k\) nondegenerate pivots, for some universal constant \(k\), with the pivot bound independent of the numerical magnitudes of \(A,b,c\)? The rule may use the complete current LP data and current basis, but may not use an oracle for an optimal basis. A positive answer establishes a polynomial worst-case simplex pivot rule; any claim of a full strongly-polynomial LP algorithm additionally requires separate verification of the arithmetic-cost model.

Let \(\mathcal X=[0,1]^D\), let \(f:\mathcal X\to[0,1]\) be deterministic and noiseless, and let a randomized sequential black-box policy observe \(Y_t=f(X_t)\), with history \(H_t=((X_1,Y_1),\dots,(X_t,Y_t))\), choose \(X_{t+1}\sim\pi_{t+1}(\cdot\mid H_t)\), and output \(\widehat X_T\sim\pi_{\rm out}(\cdot\mid H_T)\). Write \(f^\star=\inf_x f(x)\) and \(r_T(f,\pi)=\mathbb E_\pi[f(\widehat X_T)-f^\star]\). Fix the standard dyadic partition \(\mathcal P_h\) of \([0,1]^D\) into \(2^{Dh}\) cells of side \(2^{-h}\). For \(\alpha\in[\alpha_-,\alpha_+]\subset(0,1]\) and \(d\in[d_-,d_+]\subset(0,D]\), define \(\mathcal F_{\alpha,d}^{\rm up}\) by (i) \(\operatorname{osc}(f;P)\le\nu2^{-\alpha h}\) for every depth-\(h\) dyadic cell \(P\), and (ii) \(N_h(f)=\#\{P\in\mathcal P_h:\inf_{x\in P}f(x)\le f^\star+c\nu2^{-\alpha h}\}\le C2^{dh}\) for all \(h\). Let \(\mathcal F_{\alpha,d}^{\Delta}=\{f\in\mathcal F_{\alpha,d}^{\rm up}:\operatorname{osc}(f;\mathcal X)\ge\Delta\}\), where \(0<\Delta<1\). The theorem must prove that the corresponding oracle minimax risk \(R_T^\star(\alpha,d)=\inf_{\pi_{\alpha,d}}\sup_{f\in\mathcal F_{\alpha,d}^{\Delta}}r_T(f,\pi_{\alpha,d})\) is positive for the relevant finite \(T\) and determine its exact order rather than assume it. For the universal parameter set \(\Theta=[\alpha_-,\alpha_+]\times[d_-,d_+]\), define \(\Pi_{\rm univ}(\Theta)\) as policies knowing \(D,\mathcal P,\nu,C,c,\Delta,\Theta\) but not the true \((\alpha,d)\), and define \(R_T^{\rm univ}(\alpha,d;\pi)=\sup_{f\in\mathcal F_{\alpha,d}^{\Delta}}r_T(f,\pi)\). Determine the exact asymptotic behavior of \(\Gamma_T^\star(\Theta)=\inf_{\pi\in\Pi_{\rm univ}(\Theta)}\sup_{(\alpha,d)\in\Theta}R_T^{\rm univ}(\alpha,d;\pi)/R_T^\star(\alpha,d)\), including whether it is bounded, polylogarithmic, polynomial, or exhibits a phase diagram. Establish matching upper and lower bounds. The lower bound must use an explicit hard subclass \(\mathcal F_{\alpha,d}^{\rm hard}\subseteq\mathcal F_{\alpha,d}^{\Delta}\) built from multiscale packings: for sufficiently large \(h\) in a sequence \(\mathcal H\), at least \(c_0 2^{dh}\) disjoint dyadic cells can support local near-optimal alternatives of amplitude \(\delta_h\asymp\nu2^{-\alpha h}\); the optimum may lie in any packed candidate cell; evaluations outside the active cell are insufficient to identify it; and the packing yields the minimax lower bound. The proof should construct two or more regimes with early-transcript indistinguishability and conflicting oracle allocations, using a randomized distribution over deterministic hard functions only as a minimax proof device. The target is the exact oracle-relative minimax adaptation price, not merely existence of an adaptive algorithm. A second-stage extension may add unknown effective dimension and orientation, and a third-stage extension may study arbitrary tuples of geometric structure, but these are outside the first-stage theorem.

Let \(\mathcal X=[0,1]^d\), \(d\ge1\), and let \(\Sigma_L\) be the class of \(L\)-Lipschitz functions \(f:\mathcal X\to[0,1]\). At round \(t\), an adaptive policy chooses an absolutely continuous probability measure \(\mu_t(\cdot\mid\mathcal H_{t-1})\) on \(\mathcal X\), draws \(X_t\sim\mu_t\), and observes \(Y_t=f(X_t)+\eta_t\), where \(\eta_t\) is conditionally independent, mean-zero, and sub-Gaussian with variance proxy \(\sigma^2\). Let \(x^*\in\arg\min_{x\in\mathcal X}f(x)\) and \(R_T=\sum_{t=1}^T[f(X_t)-f(x^*)]\). For each \(t\), define the conditional differential entropy \(h_t=-\int_{\mathcal X}p_t(x\mid\mathcal H_{t-1})\log p_t(x\mid\mathcal H_{t-1})\,dx\). For an exploration budget \(B_T\le0\), define \(\Pi(B_T)=\{\pi:\mathbb E_\pi[\sum_{t=1}^T h_t]\ge B_T\}\). Determine the minimax regret \(\mathcal R_T^*(B_T)=\inf_{\pi\in\Pi(B_T)}\sup_{f\in\Sigma_L}\mathbb E_\pi[R_T]\). Determine whether there is a sharp critical scale for the entropy budget separating the unconstrained minimax regime from a forced-exploration regime, and characterize the exact penalty induced by the constraint. Determine whether Shannon differential entropy alone can characterize the exploration needed for minimax optimization, or whether a geometric or information-theoretic correction such as mutual information with the optimizer is necessary. Any proposed rate must use the cumulative budget or its normalized average consistently, and must not assume an unproved exponential penalty form.

Let $p,d\in\mathbb{N}$ be positive integers with $\gcd(p,d)=1$, and let $N=\max\{p,d\}>1$. Let $e=\exp(1)$ be Euler's number. Conjecture that there exists a positive integer $k$ such that $k<e(\ln N)^2$ and $pk+d$ is prime. Equivalently, every primitive arithmetic progression $d,d+p,d+2p,\ldots$ contains a prime at some positive index $k<e(\ln N)^2$. A natural weaker target is that there exists an absolute constant $C>0$ such that the same conclusion holds with $C(\ln N)^2$ in place of $e(\ln N)^2$.

Let $p$ be an odd integer with $p>69$. Define $T(p)$ to be the set of representations of $p$ as a sum of three distinct primes in strictly increasing order, and define $P(p)$ to be the set of primes that occur in at least one representation in $T(p)$. The Sum Cover Conjecture (SCC) asserts that for every odd integer $p>69$, $P(p)$ contains every odd prime $q$ satisfying $3\le q<p-6$. Equivalently, for every such pair $(p,q)$, there exist distinct odd primes $r<s$, with $r,s\ne q$, such that $r+s=p-q$.

Fix \(0<\alpha<1\), \(\nu>0\), and \(0<\gamma<1\). On \(\mathbb R^3\), consider the unforced incompressible equations \[ \partial_tu+(u\cdot\nabla)u+\nabla p+\nu(-\Delta_3)^\alpha u=0,\qquad \nabla\cdot u=0,\qquad u(0)=u_0. \] Here \((-\Delta_3)^\alpha\) is the componentwise isotropic Fourier multiplier \(|\xi|^{2\alpha}\). Suppose \(u_0\in C_b^{1,\gamma}(\mathbb R^3;\mathbb R^3)\cap L^2(\mathbb R^3;\mathbb R^3)\) is divergence free, axisymmetric without swirl, and \(\omega_0=\nabla\times u_0\) has compact support. With \(r=(x_1^2+x_2^2)^{1/2}\), this means \(u_0=u_0^r(r,z)e_r+u_0^z(r,z)e_z\); write \(\omega=\omega^\theta e_\theta\) and \(h=\omega^\theta/r\) away from the axis. An approximable strong solution on \([0,T]\) is a divergence-free distributional solution with \[ u\in C([0,T];L^2)\cap L^\infty(0,T;H^1\cap C_b^{1,\gamma}) \cap L^2(0,T;\dot H^\alpha\cap\dot H^{1+\alpha}),\qquad \partial_tu\in L^2(0,T;H^{-1}), \] and \(u\in C([0,T];C_b^{1,\gamma'})\) for every \(0<\gamma'<\gamma\). Its pressure is \(p=\sum_{i,j}R_iR_j(u_i u_j)\), modulo a function of time, where \(R_i\) is the Riesz transform with multiplier \(-i\xi_i/|\xi|\). For this interval \([0,T]\), there must exist a sequence \(\varepsilon_n>0\) decreasing to zero and solutions \(u_n\), each defined on the entire interval \([0,T]\), of the same unforced equation with the same \(\alpha,\nu\) and initial data \(u_n(0)=e^{\varepsilon_n\Delta_3}u_0\). Each \(u_n\) is divergence free, axisymmetric without swirl, and satisfies \(u_n\in C([0,T];H^m)\cap C^1([0,T];H^{m-2})\) for every integer \(m\ge3\), with the same Riesz-transform pressure convention. These all-order Sobolev conditions specify the smooth approximants and their spatial integrability; no compact-support condition is imposed on their vorticity. The single chosen sequence must converge to \(u\) in \(C([0,T];L^2)\) and in every \(C([0,T];C_b^{1,\gamma'})\), \(0<\gamma'<\gamma\). The definition requires existence of such a sequence on each finite interval, not a universal quantifier over sequences. All spatial spaces in this definition are over \(\mathbb R^3\). Is it true that every such datum has a unique global approximable strong solution, in this precise class on every finite interval, and that \[ \|u(t)\|_2^2+2\nu\int_0^t\|(-\Delta_3)^{\alpha/2}u(s)\|_2^2\,ds=\|u_0\|_2^2\quad(t\ge0), \] \[ \|h(t)\|_{L^2(\mathbb R^3)}\le\|h(0)\|_{L^2(\mathbb R^3)}\quad(t\ge0),\qquad \sup_{t\ge t_0}\|h(t)\|_\infty\le\|h(t_0)\|_\infty<\infty\quad(t_0>0)? \] No assumption \(h(0)\in L^\infty\) is imposed. Norms of \(h\) use three-dimensional Lebesgue measure, using the off-axis representative extended by zero on the axis; changing values on this measure-zero set does not change these norms. Uniqueness is within the defined approximable strong class. The statement requires neither continuity in the full \(C_b^{1,\gamma}\) norm nor an additional positive-time \(C^\infty\) assertion.

Let \(X\) be a compact metric search space and let an unknown objective \(f:X\to\mathbb R\) be accessed sequentially through either noiseless evaluations \(Y_t=f(X_t)\) or conditionally sub-Gaussian noisy evaluations \(Y_t=f(X_t)+\xi_t\). A sequential black-box optimization policy chooses each query from the history of previous queries and observations and returns a final recommendation \(\widehat X_T\) after \(T\) evaluations. Define simple regret by \(r_T=f(\widehat X_T)-\inf_{x\in X}f(x)\) for minimization. Consider a broad family of function classes indexed by local/global regularity, effective near-optimality geometry, metric entropy, and observation-noise level; this family must contain, as special cases, Hölder/Lipschitz classes with near-optimality dimension, locally smooth classes, and standard RKHS/kernel classes for which minimax black-box optimization rates are known. Determine the minimax-optimal adaptation frontier: characterize, up to universal or explicitly dimensioned constants and unavoidable logarithmic factors, the smallest penalty \(A_T(\mathcal F)\) achievable by one parameter-free policy simultaneously over every class \(\mathcal F\) in the family, relative to the oracle minimax policy that knows the class parameters in advance. The theorem must identify when simultaneous adaptation to regularity, effective dimension/near-optimality geometry, metric entropy, and noise is possible, when it is impossible, and the sharp lower bound on the price of adaptation. The result must cover both noiseless and noisy observations, arbitrary compact metric domains including high-dimensional Euclidean domains as special cases, and both global and local regularity. It should additionally determine whether there exists a canonical exploration–exploitation functional or allocation law whose induced regret is minimax-optimal throughout the admissible family, and prove an impossibility result if no such universal law exists. A complete solution must provide matching upper and lower bounds, reduce known special cases to the general theorem, and state the exact assumptions under which each adaptation claim holds.

Let \(N\ge 20\) be an even integer. Conjecture that there exist primes \(p,q\) such that \(N=p+q\), with the ordered residue pair \((p\bmod 10,q\bmod 10)\) satisfying: \(N\equiv0\pmod{10}\Rightarrow(p,q)\equiv(3,7)\pmod{10}\); \(N\equiv2\pmod{10}\Rightarrow(p,q)\equiv(3,9)\pmod{10}\); \(N\equiv4\pmod{10}\Rightarrow(p,q)\equiv(3,1)\pmod{10}\); \(N\equiv6\pmod{10}\Rightarrow(p,q)\equiv(9,7)\pmod{10}\) or \((3,3)\pmod{10}\); and \(N\equiv8\pmod{10}\Rightarrow(p,q)\equiv(1,7)\pmod{10}\). Equivalently, every even \(N\ge20\) has a Goldbach representation belonging to the designated residue-class pair prescribed by \(N\bmod10\).

This embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(μ)+\mathcal H_{p,\mathcal D}(μ)<\infty, $$ where $\mathcal C_{p,\mathcal D}(μ)$ and $\mathcal H_{p,\mathcal D}(μ)$ denote the packing energy and the Haar energy of $μ$, respectively.

Let \(Z(t)=e^{i\theta(t)}\zeta(1/2+it)\). For each simple real zero \(\gamma\) of Hardy \(Z\), define the leading local unfolding \(\rho_\zeta(\gamma)=\frac{1}{2\pi}\log\frac{\gamma}{2\pi}\) and the normalized local profile \(f_\gamma^\zeta(u)=\frac{\rho_\zeta(\gamma)}{Z'(\gamma)}Z(\gamma+u/\rho_\zeta(\gamma))\). Then \(f_\gamma^\zeta(0)=0\) and \((f_\gamma^\zeta)'(0)=1\). Write \(f_\gamma^\zeta(u)=u+\sum_{k\ge2}a_k^\zeta(\gamma)u^k\), and for fixed \(m\ge6\) set \(A_m^\zeta(\gamma)=(a_2^\zeta(\gamma),\ldots,a_m^\zeta(\gamma))\). For an explicit block of simple critical-line zeros near height \(T\), coordinate-standardize the jet vectors and let \(\Sigma_{T,m}^\zeta\) be their correlation matrix, assuming nonzero coordinate variances. For \(U_N\) Haar-distributed on \(U(N)\), with eigenangles \(\theta_1,\ldots,\theta_N\) and \(\rho_N=N/(2\pi)\), use the real secular function \(Z_U(\theta)\propto\prod_{j=1}^N2\sin((\theta-\theta_j)/2)\). At each eigenangle \(\theta_j\), define \(f_{j,N}^{\mathrm{CUE}}(u)=\frac{\rho_N}{Z_U'(\theta_j)}Z_U(\theta_j+u/\rho_N)=u+\sum_{k\ge2}a_{k,j}^{\mathrm{CUE}}u^k\), form \(A_m^{\mathrm{CUE}}=(a_2^{\mathrm{CUE}},\ldots,a_m^{\mathrm{CUE}})\), coordinate-standardize, and let \(\Sigma_{N,m}^{\mathrm{CUE}}\) be the resulting correlation matrix. As a generic analytic control, take \(f^{\mathrm{Taylor}}(u)=u+\sum_{k=2}^m\xi_k u^k\) with iid centered unit-variance coefficients and apply the same standardization and PCA. Primary problem: for each fixed \(m\ge6\), determine whether suitable limits \(\Sigma_{T,m}^\zeta\to\Sigma_m^\zeta\) as \(T\to\infty\) and \(\Sigma_{N,m}^{\mathrm{CUE}}\to\Sigma_m^{\mathrm{CUE}}\) as \(N\to\infty\) exist under explicit sampling/block conventions, and whether \(\Sigma_m^\zeta=\Sigma_m^{\mathrm{CUE}}\), or at least whether their leading eigenspaces and normalized eigenvalue spectra agree asymptotically. For \(m=6\), resolve the observed leading two-dimensional structure. Structural subproblem: explain or refute the experimentally observed even/odd organization of the jet coefficients. After factoring the forced simple zero, study \(\log(f_\gamma(u)/u)=\sum_{k\ge1}c_k(\gamma)u^k\). Derive, with all canonical-product or regularization terms made precise, exact or asymptotic relations between the \(c_k\) (and hence the \(a_k\)) and signed local zero/eigenangle statistics. Determine whether the parity split reflects a symmetric/antisymmetric decomposition of the local environment, whether the apparent two-dimensionality is generic, CUE-universal, or instead a CUE background plus a persistent zeta-specific arithmetic correction. A rigorous treatment should also test robustness under the intrinsic unfolding \(\rho_*(\gamma)=\theta'(\gamma)/\pi\). Negative resolutions are allowed, including failure of convergence, disappearance of the spectral gap, or a null mechanism explaining the PCA structure.