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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.
The sharp Hlawka inequality for Schatten -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
Markov decision processes assume that the transition law is known exactly. In practice it is estimated from data, and the optimal policy computed for the point estimate can perform badly when the true law differs. Robust dynamic programming replaces each transition law by a set of plausible laws and evaluates a policy by its worst-case expected reward over that set. The worst case is taken against an adversary (nature) who may choose a law from the set at every step.
For this worst case to be computable by backward induction, the set of path measures of a policy has to decompose epoch by epoch. Garud Iyengar named this property Rectangularity and proved that under it the classical finite horizon theory carries over: a robust Bellman equation holds, and deterministic Markov policies are optimal among all history dependent randomized policies, with state and action sets that may be countably infinite and ambiguity sets that need not be convex (Iyengar, CORC Tech Report TR-2002-07, rev. 2004; published in Math. Oper. Res. 30(2), 2005).
Timeline:
A finite horizon ambiguous Markov decision process (AMDP) has decision epochs , , and a terminal epoch . States lie in a countable set and actions in a countable set . In state at epoch the admissible actions form a nonempty set . For each admissible action , the next state is drawn from some probability measure , where is a nonempty set of probability measures on , the ambiguity set. The decision maker receives when is taken in and the next state is , and at the terminal epoch.
A history at epoch is . A policy assigns to each history a probability measure on . It is deterministic () if every is a point mass, and deterministic Markov () if moreover the chosen action depends on the current state alone. denotes all history dependent randomized policies.
Under Rectangularity (Assumption 1), the set of path measures consistent with is a product of one-epoch sets: nature chooses, for every epoch , every history and every admissible action , a measure , and the path measure is . The robust value of from and the robust value function are
The optimistic values , replace the infimum over by a supremum.
For , depends on only through ; for , ; and
The mission also contains, as a draft theorem that is not a milestone, the per-policy identity of the proof of Theorem 1, eq. (12): for each policy, .
Theorem 2 is the basis of robust dynamic programming in finite horizon: computing an optimal robust policy reduces to backward passes, each a family of one-stage problems . It also justifies restricting attention to deterministic Markov policies, which is what every algorithm in the robust MDP literature computes. Theorem 1 isolates the role of Rectangularity: without it the worst case over path measures does not decompose and the recursion characterizes nothing.
The results are proved on paper; no machine-checked proof is known. The platform holds the finite-state, Markov-controller special case (the Nilim–El Ghaoui RobustMDP series, posed, not proved), whose comparison class is too small to express the claim suffices against . A formal proof here supplies the history dependent policy and adversary machinery on countable spaces that the discounted robust theory and the robust MDP literature reuse.
The obvious argument writes the robust value as an inf–sup over path measures and pushes the infimum inside the expectation epoch by epoch. That step is the content of (12): the infimum of an expectation equals the expectation of an infimum only because nature's continuation choice may depend on the realized action and next state, so near-worst continuations for different branches can be glued into one admissible adversary. Infima need not be attained (the ambiguity sets are arbitrary, the state set countable), so the gluing is an -argument over countably many branches. The supremum over policies needs a matching splicing of -optimal continuation policies (13)–(14). Finally, randomized decision rules must be eliminated (15), which uses Rectangularity again, now per action. Restricting either the policies or the adversary to Markov ones from the start is not allowed: it changes the comparison class and makes the goal circular.
S and one countable action type A serve all epochs; the page's epoch-dependent sets embed into their disjoint union. Epochs are 0-based natural numbers, with terminal epoch M.N and 1 ≤ N.(Fin n → S × A) × S. Probability measures are PMF; as a tsum. The expected reward under a policy and an adversary is computed by backward iterated sums, which is the expectation under the product path measure (3).t < N; the supremum over equals that over the page's because uses only . An adversary chooses for every epoch, history and admissible action.Definitions: AMDP, History, History.extend, IsPolicy, Policy, IsDeterministic, IsDetMarkov, Adversary, Selection, expect, expTail, expectedReward, V, Vstar, Vbar, VbarStar, all in RobustDP.FiniteHorizon. Lemmas about gluing adversaries along history prefixes and about suprema of bounded families indexed by subtypes are reusable. Contributions to any milestone are welcome, in any order; (15) is a one-epoch statement independent of the rest.
A mathematical program with complementarity constraints (MPEC, also called MPCC) is a nonlinear program in which some pairs of constraint functions must be nonnegative with at least one of each pair equal to zero. Such programs model bilevel optimization, Stackelberg games, contact problems in mechanics, and traffic equilibrium design; see Luo, Pang and Ralph, Mathematical Programs with Equilibrium Constraints (Cambridge University Press, 1996). The complementarity constraints make every standard constraint qualification of nonlinear programming fail at every feasible point: neither LICQ nor MFCQ ever holds. Off-the-shelf NLP theory and solvers therefore do not apply directly.
Relaxation methods work around this by replacing the MPEC with a family of ordinary nonlinear programs indexed by a parameter and letting . The oldest of these is the global relaxation of Scholtes (SIAM J. Optim. 11, 2001). The relaxed programs are only useful if they are themselves well posed: a local minimizer of a relaxed program should carry Lagrange multipliers, so that the sequence of KKT points the method computes actually exists. Hoheisel, Kanzow and Schwartz (Preprint 299, Univ. Würzburg, 2010; published in Mathematical Programming) compare five relaxation schemes, prove convergence under weaker MPEC constraint qualifications than before, and ask which standard constraint qualification the relaxed programs satisfy. This mission formalizes their answer for Scholtes' scheme, Theorem 3.2.
A standard nonlinear program (2) on minimizes subject to and for finitely many and . At a feasible , the active set is . A family is positive-linearly dependent if some combination vanishes with and not all coefficients zero. The Mangasarian–Fromovitz constraint qualification (MFCQ) holds at if the gradients are linearly independent and some direction has for every active and for every .
The MPEC (1) minimizes subject to
with continuously differentiable data. At a feasible point the indices of the complementarity pairs split into (), () and (). The tightened program TNLP replaces each pair by equalities on the components that vanish at and a sign constraint on the other. MPEC-MFCQ holds at if standard MFCQ holds for TNLP at .
Scholtes' relaxed program , for , keeps , , , and replaces by . Its feasible set is . Lean notation: the MPEC is MPEC n m p l, TNLP is P.TNLP xs, is P.RS t, and MFCQ of an NLP Q at x is Q.IsMFCQ x.
If is feasible for the MPEC and MPEC-MFCQ holds at , there are a neighbourhood of and such that
The neighbourhood is fixed before and works for every .
Theorem 3.2 says that the relaxed programs inherit a standard constraint qualification from the MPEC. Consequently every local minimizer of near is a KKT point, which is exactly the hypothesis of the convergence theorem for Scholtes' method (Theorem 3.1 of the paper: limits of KKT points of are C-stationary under MPEC-MFCQ). Without it, the convergence theorem could be about sequences that do not exist. The result also replaces the MPEC-LICQ-based regularity results of earlier work by the weaker MPEC-MFCQ.
The theorem is proved in the paper; to our knowledge no part of this theory has been machine-checked. A formal development delivers a reusable layer for nonlinear programming in Lean: positive-linear dependence, MFCQ and its dual characterization, the tightened program of an MPEC and the MPEC constraint qualifications. Sibling missions of this series formalize Theorem 3.1 and the Kadrani–Dussault–Benchakroun relaxation (Theorem 3.5) on the same vocabulary.
The obvious argument is a continuity argument: MFCQ is an open condition, so it should persist near . This fails as stated, because MFCQ for is not a perturbation of MFCQ for TNLP: the two programs have different constraints, and the active set of at changes with and . Near a biactive index , the product constraint can be active, and its gradient tends to zero as . So it cannot be treated as a small perturbation of any gradient in the TNLP family. In addition, the neighbourhood must be uniform in , while the active product constraints depend on . The step that needs care is the regrouping of the multiplier equation of into a combination of TNLP-type gradients, with every coefficient's sign accounted for. The separate step from MPEC-MFCQ to positive-linear independence needs a theorem of the alternative (Motzkin's transposition theorem), which is not in Mathlib in this form.
EuclideanSpace ℝ (Fin n); is the inner product of gradient f x with d. Indices are Fin m (0-based), and are allowed.P.IsC1 on the goal and on milestones 3–5. Milestones 1–2 do not need it.𝓝 xs chosen before .The goal would be trivialized by a weakened MFCQ, for instance one that omits the linear independence of the equality gradients or requires the strict inequality only for some of the active constraints. Such a formalization is ruled out: IsMFCQ requires both conditions, over the full active set of . The neighbourhood must be a genuine element of 𝓝 xs, so an empty is excluded.
Welcome contributions: a proof of Remark 2.2, general lemmas on persistence of positive-linear independence under continuous perturbation, and the final assembly. These lemmas apply to nonlinear programming in general, not only to this mission.
A force whose derivatives decay rapidly can still encode a computation in a particle trajectory. The selected construction makes the observer condition and effectiveness requirements precise. The pinned manuscript supplies the research context.
For each finite deterministic machine and valid input, compilers produce smooth force and velocity fields on nonnegative time and real three-space, with common compact spatial support.
The selected goal is OAI.AlternatingNS.alternating. Its central assertion is
The theorem states that for every positive real viscosity ν computable by rational approximations with error at most 2⁻ⁿ, there exist computable compilers for a force f and velocity U, and one compact set K⊂ℝ³, with the following property for every well-formed finite deterministic Turing machine M and valid finite input w. The compiled programs describe smooth fields f,U on [0,∞)×ℝ³, both spatially supported in K for all nonnegative times, such that every mixed space-time derivative D satisfies sup_{t≥0,x}(1+t+‖x‖)ᴶ‖Df(t,x)‖<∞ and the analogous bound for U, for every nonnegative integer J. Each program supplies rational approximations to all mixed derivatives at rational space-time points with requested error 2⁻ᵏ, effective moduli of continuity on bounded regions, and integer bounds for all these weighted derivative norms. The velocity starts from zero, is divergence-free, and solves the classical forced Navier–Stokes equation ∂ₜU+(U·∇)U=νΔU+f with identically zero pressure; specifically f=∂ₜU+(U·∇)U−νΔU. On every interval [0,T], U is continuous in H² and continuously differentiable in L², and U and its spatial derivative are uniformly bounded in space and time. Moreover, any classical solution (v,p) with the same force and zero initial velocity equals (U,0) pointwise for all nonnegative times, provided on every [0,T] it has v continuous in H², continuously differentiable in L², and uniformly bounded, and p continuous in H¹. Here these Sobolev continuity conditions mean continuous L² representatives of every spatial derivative through the indicated order; time derivatives at zero are taken within [0,∞). Every initial point a has a unique trajectory X(a,t) for t≥0 satisfying X(a,0)=a and ∂ₜX(a,t)=U(t,X(a,t)). Finally, the trajectory starting at (−1,0,0) has strictly positive first coordinate at some nonnegative time if and only if M halts on w, where encountering a missing instruction also counts as halting.
The goal is the alternating-coordinate construction at every positive computable viscosity. It includes Sobolev time regularity, effective bounds, zero initial velocity and the exact observer condition, rather than all three manuscript constructions. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The encoding must preserve all weighted derivative bounds and uniqueness in the stated comparison class while recording arbitrarily long computations.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Incompressible transport can implement prescribed geometric operations on coding sheets. The selected statement concerns one such finite transport construction with effective smooth forcing. The pinned manuscript supplies the research context.
Source and target rectangles have rational data and specified positive diagonal ratios. In the formula, D_i(y)=q_i+diag(r_i)(y−p_i), with source and target centers p_i and q_i. Space is represented periodically with period ten, and the viscosity is a positive computable real.
The selected goal is OAI.Solenoidal.sheet_theorem. Its central assertion is
The theorem states that, for any natural number N, any sheet datum D of size N, and any computable real viscosity ν>0, there exist a forcing field f and a velocity field u (each a time-dependent vector field on ℝ³) and a flow map X such that the following hold. Here D consists of N source rectangles and N target rectangles with rational corners, all inside the square [2,3]², with the sources pairwise separated by a positive distance and likewise the targets, together with positive rational ratios in each of two coordinates such that the diagonal affine map from source i to target i, which sends the source center to the target center and scales offsets by the ratios, maps the source rectangle exactly onto the target rectangle. Both f and u are periodic in space with period 10 in every coordinate and are C^∞ in space and time jointly. The forcing f is effective, meaning that all its mixed space-time partial derivatives can be approximated to any rational accuracy by a single fixed partial recursive procedure from computable names of the evaluation point. Moreover f has zero mean over the fundamental cell [0,10)³ at every time, is divergence free, and is 1-periodic in time. The field u is a classical solution on t≥0 of the forced Navier–Stokes equations ∂ₜu + (u·∇)u = −∇p + ν Δu + f with zero pressure, zero initial data, incompressibility and spatial periodicity, so u is a solution driven by f without pressure. Any classical solution v with pressure p for the same f and ν and with zero initial velocity coincides with u, and has p identically zero, for all t≥0. The flow map X satisfies X(0,a)=a and solves the particle-path equation dX/dt=u(t,X) for t≥0, and for each i and each point y of the i-th source rectangle, the time-1 flow image of the sheet point (y₀,y₁,2) agrees modulo the torus ℝ³/(10ℤ)³ with the sheet point at the diagonal-map image of y. Further, u vanishes in a neighborhood of every integer time, the advection term (u·∇)u vanishes everywhere, and both f and u have all their mixed space-time derivatives uniformly bounded by rational bounds computable from the derivative multi-index by a fixed partial recursive procedure. Finally, if N=0 then f and u are identically zero.
The target is the sheet theorem, including its encoded uniqueness statement and zero-advection conditions. Machine-halting detection is not a conclusion of this published goal. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The construction must simultaneously enforce smoothness, incompressibility, temporal periodicity, effective bounds and exact transport on every point of each sheet rectangle.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Homogeneous minimizers model possible free-boundary singularities. A nonflat example in seven dimensions supplies one side of the claimed critical-dimension threshold. The pinned manuscript supplies the research context.
The one-phase Bernoulli energy on a ball is the integral of squared weak gradient plus the indicator of positivity. Competitors are nonnegative Sobolev functions with the prescribed H1-zero-boundary difference.
The selected goal is OAI.Bernoulli.nonflat_in_seven. Its central assertion is
The theorem states that there exists a function u: ℝ⁷ → ℝ that is nonnegative almost everywhere, belongs locally to H¹, and is a global minimizer of the one-phase Bernoulli energy E_B(u) = ∫B (|∇u|² + 1{u>0}) dx. Here ∇u is a distributional gradient, and global minimality means that, for every open ball B of positive radius and every nonnegative almost-everywhere Sobolev competitor v ∈ H¹(B) with v − u ∈ H¹₀(B), one has E_B(u) ≤ E_B(v). Membership in H¹₀(B) requires approximation by smooth functions compactly supported in B, with both the functions and their gradients converging in L². The function is nonzero in the sense that it is not equal to zero almost everywhere, and it is homogeneous of degree one: for each real r > 0, u(rx) = r u(x) for almost every x. Nevertheless, it is not flat: there is no unit vector e ∈ ℝ⁷ for which u(x) = max(⟨x,e⟩, 0) almost everywhere. All almost-everywhere statements and integrals use Lebesgue measure.
The selected goal is existence of a nonzero nonflat one-homogeneous global minimizer in dimension seven. Flatness in dimensions at most six and singular-set dimension bounds are not attached conclusions. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
A stationary cone is insufficient: the example must minimize against all admissible ball competitors and fail every half-space profile.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Negative curvature suggests strong analytic control, but bounded holomorphic coordinates impose a different global condition. The target constructs a complex threefold separating them. The pinned manuscript supplies the research context.
The construction uses the encoded Hartogs domain and a complete Kähler metric with sectional curvature between two negative constants. Holomorphic coordinate obstructions are stated in the published main proposition.
The selected goal is OAI.PinchedHartogs.main_theorem. Its central assertion is
The theorem states that there exist a subset M of ℂ³ (realized as ℂ² × ℂ) and a matrix-valued function g assigning a 3×3 complex matrix to each point, together with real numbers A and B, such that M is open, nonempty and contractible, and g is a Kähler metric on M, meaning g is C^∞ on M, Hermitian at each point (g_ij = conj(g_ji)), positive definite (the Hermitian form of g at u has positive real part for every nonzero u), and satisfies the closedness condition ∂g_kj/∂z_i = ∂g_ij/∂z_k, with ∂ the Wirtinger derivative (1/2)(∂_x − i∂_y). Moreover the metric is geodesically complete on M: for every point p in M and every vector v there is a C^∞ curve γ: ℝ → M with γ(0)=p, γ'(0)=v that satisfies the geodesic equation γ''k + Σ Γ^k{ac} γ'_a γ'_c = 0 for all t, where the Christoffel symbols are built from the inverse of g and the ∂ derivatives of g. The metric is also negatively pinched: 0 < A ≤ B and, for every point of M and every real-linearly independent pair u, v, the sectional curvature, computed from the explicit curvature tensor formula in the source, lies between −B and −A. In addition, M has no bounded coordinates: there is no map F from ℂ³ to ℂ³, holomorphic on a neighbourhood of every point of M and bounded in every component on M, whose complex Jacobian determinant is nonzero at every point of M. Finally, M is not biholomorphic to a bounded domain, that is, there is no bounded open set N with holomorphic maps F on M and G on N, each analytic near the respective set, mapping M into N and N into M, which are mutually inverse.
The selected main theorem requires a nonempty open contractible domain, a geodesically complete Kähler metric with two-sided negative real-sectional pinching, no bounded holomorphic map to complex three-space with everywhere-nonsingular differential, and no biholomorphism to a bounded domain. The controlled potential and metric-transfer statements are attached as supporting milestones. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
Metric completeness and two-sided negative curvature must coexist with a global obstruction to bounded holomorphic coordinates.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
adc7f1241b42. Selected goal source.Affine maximal graphs satisfy a nonlinear fourth-order geometric equation. The target asks whether completeness forces such a locally uniformly convex graph to be a paraboloid. The pinned manuscript supplies the research context.
The dimension is between three and nine. The domain is nonempty, open and convex, the function is smooth with positive-definite Hessian, and completeness uses intrinsic Euclidean graph length.
The selected goal is OAI.AffineBernstein.affine_bernstein. Its central assertion is
The theorem states that, for every integer dimension 3 ≤ n ≤ 9, a nonempty open convex set Ω ⊆ ℝⁿ and a function u : ℝⁿ → ℝ that is smooth on Ω satisfy the following conclusion under three hypotheses. First, the Hessian H = D²u is positive definite at every point of Ω. Second, u satisfies the affine maximal equation ∑ᵢⱼ Uᵢⱼ ∂ᵢ∂ⱼw = 0 throughout Ω, where U = (det H)H⁻¹ and w = (det H)^{−(n+1)/(n+2)}. Third, the graph is complete for its intrinsic Euclidean path distance: the distance between x and y in Ω is the infimum over C¹ paths γ in Ω joining them of ∫₀¹ √(‖γ′(t)‖² + (Du(γ(t))γ′(t))²) dt, and every Cauchy sequence for this distance has a limit in Ω for the same distance. Then Ω = ℝⁿ and there exist a symmetric positive definite real n × n matrix A, a vector b ∈ ℝⁿ, and c ∈ ℝ such that u(x) = ½xᵀAx + b·x + c for every x ∈ ℝⁿ. Moreover, an invertible affine map of ℝⁿ × ℝ carries the standard paraboloid {(x, ‖x‖²)} onto the graph {(x, u(x)) : x ∈ Ω}.
The goal also gives an invertible affine map from the standard paraboloid to the graph. The wider immersed-hypersurface formulation is not included as a separate target. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The domain is not initially assumed to be all of space. Both global domain exhaustion and the quadratic form must follow from the equation and completeness.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Sharp filling inequalities compare the mass of a cycle with the least mass needed to fill it. The selected formulation extends a Euclidean coefficient to proper CAT(0) metric spaces. The pinned manuscript supplies the research context.
Integral currents are encoded metric currents with integer-multiplicity chart decompositions and integral boundary. The input is a compactly supported n-cycle for n at least two. In the displayed coefficient, σ_n=(n+1)ω_{n+1}, where ω_{n+1} is the Euclidean volume of the unit ball in dimension n+1; this is the published sphereArea convention.
The selected goal is OAI.CAT0Fillings.sharp_integral_filling. Its central assertion is
The theorem states that, for a proper metric space X with its Borel σ-algebra that is CAT(0), meaning there is a choice of geodesic parametrization segment(x,y,t) from x to y (with segment(x,y,0)=x, segment(x,y,1)=y, and dist(segment(x,y,s),segment(x,y,t))=|s−t|·dist(x,y) for s,t in [0,1]) satisfying the comparison inequality dist(segment(o,x,s),segment(o,y,t))² ≤ (s·d(o,x) − t·d(o,y))² + s·t·(d(x,y)² − (d(o,x) − d(o,y))²), the following filling property holds for every integer n ≥ 2. Let T be an integral n-current on X, that is, a metric current (a multilinear functional on a bounded Lipschitz function and n Lipschitz functions, with locality, continuity and finite mass) that is a countable sum of disjointly supported integer-multiplicity bi-Lipschitz chart pieces and whose boundary has the same properties, and suppose T is compactly supported and a cycle, meaning its boundary is zero. Then there exists a compactly supported integral (n+1)-current S on X whose boundary is T and whose mass satisfies mass(S) ≤ c_n · mass(T)^((n+1)/n), where c_n = 1/((n+1)·(σ_n)^(1/n)), σ_n = (n+1)·ω_{n+1} is the area of the unit n-sphere and ω_{n+1} is the volume of the unit ball in Euclidean (n+1)-space. This is an admitted theorem statement, not a verified proof.
The goal is the CAT(0) integral-current filling inequality. Euclidean coefficient optimality is an additional reference. The manifold perimeter comparison and equality-rigidity statements in the manuscript are not separate attached targets. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The sharp coefficient must hold without a smooth manifold model, while preserving integrality, compact support and the exact boundary.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
adc7f1241b42. Selected goal source.Recursive potentials can define tree norms with controlled midpoint behavior. The central question is whether that control forces an asymptotically uniformly convex renorming. The pinned manuscript supplies the research context.
The encoded constructions include root-sum and zero-root completions, finite tree heads, aggregation spaces and variable-exponent coordinate models. Two least nonnegative potential fields determine the norms.
The selected goal is OAI.ComparatorModel.RecursivePotentials.main. Its central assertion is
The theorem states (its proof is admitted with sorry) that the defined proposition MainClaim holds. MainClaim asserts that there exist constructions, over index types in universe 0, of the finite rooted tree head structure, the tree-vector spaces, the zero-root tree-vector spaces, the aggregation-vector spaces, and their l^p-coordinate models, such that eleven statements about completions of finite-support vectors on rooted trees hold simultaneously. These comprise: (1) completion statements saying that the root-sum tree space for the sequence tree and for the joined tree, and the zero-root space for the joined tree, are complete, infinite-dimensional, with continuous coordinates extending the vector coordinates, continuous potential fields P and Q extending the finite-tree ones, norm equal to P+Q at the root (or both P and Q at the root in the zero-root case, where the root coordinate vanishes), and with norm-nonincreasing idempotent coordinate projections onto initial subtrees having finite-dimensional range, closed finite-codimensional tails and approximation of tail elements by vectors vanishing on the head, and that XZero and XJoined are reflexive; (2) a main statement giving the lower bound t^3/128 on the averaged modulus of XSigma and XJoined for 0<t<1, a cubic-type uniform convexity inequality (‖x+z‖+‖x-z‖)/2 ≥ ‖x‖+‖z‖^3/(8(2‖x‖+‖z‖)^2) for x supported on a head and z vanishing there, and that no equivalent norm on XSigma, XZero or XJoined has the AUC property (positive modulus at every positive t); (3) positivity of the averaged modulus of XSigma at every positive radius t; (4) a separation statement for XJoined: for each ε>0 there is η in (0,1), equal to min(1/2, logarithmicGamma(2, ε/16)) when ε≤2, such that if ‖x±z_n‖≤1 for all n and the z_n are pairwise at distance at least ε, then ‖x‖≤1-η; (5) properties of the variable-exponent l^2-sum of height-h tree spaces with exponents 1+1/h, namely completeness, reflexivity, separable dual, a root-sum functional ν (nonnegative, definite, subadditive, absolutely homogeneous) bounded above and below by explicit multiples of the l^p norm, dense finitely supported vectors, and finite-rank projections; (6) a recursive characterization of the l^p-model potentials by least pairs, with ν equal to the root P plus Q; (7) for every equivalent norm A on the variable space, the modulus at lower/upper vanishes; (8) the absence of any equivalent AUC norm there; (9) a sixth-power stability inequality and (10) a cubic stability inequality for admissible heads, with explicit constants; and (11) a weak-tail estimate: if x and a weakly null sequence y_j with ‖y_j‖≥ε, 0<ε≤1, satisfy ‖x±y_j‖≤1, then ‖x‖≤θ(ε)<1.
The published MainClaim is an eleven-part package, not only the displayed midpoint inequality. Its additional completion, separation, stability, weak-tail and no-AUC assertions remain part of the goal. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The bundled target must coordinate completion, continuous coordinate maps, finite-head approximations, reflexivity and uniform estimates across several related spaces.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Independent positive multipliers on a branching tree generate concrete subspaces of real L1. Their asymptotic geometry can distinguish midpoint estimates from AUC renormability. The pinned manuscript supplies the research context.
A nonconstant positive multiplier has probability law μ, mean one and finite second moment. Path products are multiplied along nonempty prefixes, and their L1 classes generate a closed real span.
The selected goal is OAI.IndependentProducts.main_general. Its central assertion is
The theorem states that, for any Borel measure μ on ℝ satisfying the multiplier hypotheses, the closed real L¹ span of the path products is infinite-dimensional, and no equivalent norm on it is AUC. The hypotheses say that μ is a probability measure, almost every value is positive, μ is not almost surely equal to any constant, its mean is 1, and the identity function has a finite second moment (is in L²). Vertices are finite lists of positive integers, a sample assigns a real number to each nonroot vertex, and the product measure is the infinite product of independent copies of μ, one per nonroot vertex. The path product of a vertex v multiplies the sample values at the nonempty prefixes of v, and is 1 for the root. productSpan(μ) is the topological closure, in L¹ of the product measure, of the real linear span of the classes equal almost everywhere to some path product. The theorem concludes first that productSpan(μ) is not finite-dimensional over ℝ. Second, for every seminorm N on productSpan(μ) that is an equivalent norm, meaning there are constants 0<a≤b with a‖x‖≤N(x)≤b‖x‖ for all x, N fails the AUC property, which requires that, for every t>0, the one-sided asymptotic modulus of N at t is strictly positive. That modulus is the infimum over N-unit vectors x of the supremum over closed finite-codimension subspaces F of the infimum of N(x+ty)−1 over y in F with N(y)=1. The statement is given as an admitted theorem without proof.
The selected general target asserts infinite dimension and absence of AUC renormings. Specific exponential and Gaussian-square midpoint estimates are separate published targets, retained as additional references. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The obstruction must work for every admissible multiplier distribution and every equivalent norm, rather than only a particular exponential example.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
OAI.IndependentProducts.main_exponential (Open).OAI.IndependentProducts.main_gaussian_12 (Open).OAI.IndependentProducts.main_gaussian_24 (Open).adc7f1241b42. Selected goal source.A positive averaged midpoint modulus need not supply a positive one-sided asymptotic modulus. Tree-potential norms give explicit spaces in which to separate those notions. The pinned manuscript supplies the research context.
Coordinates are finite lists of natural numbers. Tests have bounded prefix potentials, one of the specified quadratic budgets and a budget-dependent support condition; the normed space is completed.
The selected goal is OAI.BoundedTreePotentials.TreeCalculus.main_counterexample. Its central assertion is
The theorem states that, for every choice of a Boolean flag r (whether the root node, the empty list, is included as a coordinate) and every quadratic kind k among sibling, antichain and global, the following hold for the Banach space X = TestCompletion(treeTestFamily r k). Here nodes are finite lists of natural numbers, the potential of a coefficient function f at a node s is the sum of f over all prefixes of s, and a coefficient function f is a tree test if f vanishes at the root when r is false, every potential satisfies |potential f s| ≤ 1, a quadratic budget holds, and a support condition holds. The budget says that the sum of f(s)² over every admissible finite set B of nodes is at most 1, where admissible means: all of B are children of one common parent (sibling), pairwise prefix-incomparable (antichain), or any finite set (global). The support condition is vacuous for sibling, finite support for antichain, and square-summability for global. The tree test family is the set of such functions on the coordinates, and each finitely supported vector x is given the norm sup over tests f of |Σ x_i f_i|; X is the completion of this normed space. First, X is complete, separable, and not finite-dimensional over ℝ. Second, for every 0 < t < 1, the averaged modulus of X, defined as the infimum over unit vectors x of the supremum over closed finite-codimension subspaces F of the infimum over y in F with ‖y‖ ≥ 1 of (‖x+ty‖+‖x−ty‖)/2 − 1, is at least √(1+t²/4) − 1. Third, for every real normed space Y that is linearly homeomorphic to X, Y does not satisfy IsAUCReal, meaning it is false that the one-sided modulus (defined like the averaged one but with ‖y‖ = 1 and the quantity ‖x+ty‖ − 1) is positive for every t > 0.
The formal statement quantifies over both root flags and all three encoded quadratic kinds. Its claims are completeness, separability, infinite dimension, the stated modulus bound and no linearly equivalent AUC space. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The lower estimate must coexist with an obstruction to every equivalent AUC norm, not just failure for the original tree norm.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Uniform discreteness places a lower bound on distances between points. The target tests whether that metric simplicity forces uniformly bounded finite-rank approximation in the associated Lipschitz-free space. The pinned manuscript supplies the research context.
The free space is the closed real span of point evaluations in the dual of Lipschitz functions vanishing at a base point. Approximation and bounded approximation retain their distinct operator-norm quantifiers.
The selected goal is OAI.DiscreteFree.source_endpoints. Its central assertion is
The theorem states that two propositions hold together, both about the Lipschitz-free space FreeSpace(o) of a pointed metric space (M,o), which is the closed linear span, inside the dual of the Banach space Lip0(o) of real Lipschitz functions vanishing at o (normed by the supremum of the slopes |f(x)-f(y)|/d(x,y)), of the point evaluations; delta(o,a) denotes the evaluation at a. The first, BlockStatement, says that for every natural number p ≥ 1 there exist a countable metric space M and a base point o with all distinct points at distance at least 1 and all distances at most 300p²+150p+5, together with a finite set A of M containing o, such that every bounded linear operator T on FreeSpace(o) of finite rank and norm at most p moves some delta(o,a) with a in A by at least 1/2, that is ‖T(delta(o,a))-delta(o,a)‖ ≥ 1/2. The second, MainStatement, says there exist a countable metric space M and a point o such that distinct points are at distance at least 1, M is unbounded, M is not a proper space, FreeSpace(o) has the approximation property (finite-rank bounded operators approximate the identity uniformly on every compact set to within any ε>0), and yet for every Λ ≥ 1 FreeSpace(o) fails the bounded approximation property with constant Λ, meaning the approximating finite-rank operators cannot all be chosen with norm at most Λ. The theorem is stated with sorry, so no proof is asserted here.
The selected conjunction includes quantitative block obstructions and an unbounded, nonproper, uniformly discrete example. The quantitative real-ℓ1 renorming target is retained as an additional reference. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
Compact-set approximation must remain possible while every candidate uniform norm bound fails. The finite block obstruction must survive in one countable space.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
adc7f1241b42. Selected goal source.Markov type is a metric inequality for the displacement of reversible chains. The target connects this probabilistic control to the possibility of uniformly convex renorming. The pinned manuscript supplies the research context.
Chains are finite, stationary and reversible. A map sends their states into a complete real normed space, and the inequality compares n-step and one-step pth displacement moments.
The selected goal is OAI.MarkovSuperreflexivity.hasNontrivialMarkovType_iff_hasEquivalentUCNorm. Its central assertion is
The theorem states that for a real normed space X that is complete (a Banach space), X has nontrivial Markov type if and only if X admits an equivalent uniformly convex norm. Nontrivial Markov type means there exist p>1 and K>0 such that for every finite reversible Markov chain on m states {0,…,m−1}, with stationary probability vector π, nonnegative row-stochastic transition matrix P, and detailed balance π(s)P(s,t)=π(t)P(t,s), every map f from the states to X, and every n≥1, the expected value of ‖f(Z_n)−f(Z_0)‖^p along n-step paths, started from π, is at most K^p·n times the corresponding one-step expectation of ‖f(Z_1)−f(Z_0)‖^p. Having an equivalent uniformly convex norm means there is a seminorm q on X and constants a,b>0 with a‖x‖≤q(x)≤b‖x‖ for all x, such that for every ε in (0,2] there is δ>0 with q((x+y)/2)≤1−δ whenever q(x)≤1, q(y)≤1 and q(x−y)≥ε.
The exact target uses uniformly convex norms with two-sided equivalence constants. It is not phrased as a direct uniformly smooth norm construction. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The inequality is uniform over all finite chains and state maps, while the conclusion is one norm controlling all vectors of the Banach space.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.The separable quotient problem asks whether every infinite-dimensional Banach space admits a separable infinite-dimensional quotient. The selected statement isolates a negative result under the continuum hypothesis. The pinned manuscript supplies the research context.
A quotient is encoded as a surjective bounded linear map between Banach spaces over the same scalar field. The target considers both real and complex scalars.
The selected goal is OAI.SeparableQuotient.negative_main. Its central assertion is
The theorem states that, under the continuum hypothesis CH (the defined proposition that the cardinality of ℝ equals ℵ₁), the separable quotient assertion fails over both the real and the complex numbers, at any universe level u. Here SQ over a scalar field 𝕜 (ℝ or ℂ) asserts that every complete normed space X over 𝕜 in Type u that is not finite-dimensional has a separable quotient in the following sense: there exist a complete, separable, infinite-dimensional normed space Y over 𝕜, also in Type u, and a bounded 𝕜-linear map T from X to Y that is surjective. The conclusion is the conjunction of ¬SQ(ℝ) and ¬SQ(ℂ), so for each field there is an infinite-dimensional Banach space with no such bounded surjection onto an infinite-dimensional separable Banach space. The statement is admitted without proof in the source.
CH is an explicit assumption. The published goal does not state relative consistency, a positive model, or the complete independence assertion in the manuscript title. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The obstruction must rule out every bounded surjection onto every separable infinite-dimensional target space, rather than one proposed quotient.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.The homotopy hypothesis relates algebraic higher groupoids to spaces. Elementary expansions provide a concrete test of whether adjoining a higher cell preserves homotopy information. The pinned manuscript supplies the research context.
C is a globular coherator in the encoded convention, X is a cellular model, and Y is a pushout that attaches an (n+1)-disk along the source-face inclusion of an n-disk.
The selected goal is OAI.Grothendieck.elementary_expansion. Its central assertion is
The theorem states that, for a coherator C (a globular theory, meaning a category of shapes built from globes by iterated gluing, with morphisms preserving the globular pushouts, which has a cellular presentation as a countable-stage colimit of free extensions along admissible pairs of parallel cells and in which every admissible pair is filled by some morphism), the following holds for models of C, i.e. presheaves on C sending globular pushouts to pullbacks. Let X be a cellular model, meaning that it is built from an initial model by a well-ordered transfinite composition in which each successor step attaches cells freely along boundaries of cells of the previous stage. Let n be a natural number, let a be a map from the free model disk(n) on the n-globe to X, and let J_n : disk(n) → disk(n+1) be the map induced by the source-face inclusion of the n-globe into the (n+1)-globe. If Y, together with i : X → Y and b : disk(n+1) → Y, forms a pushout square of a and J_n, then i is a weak equivalence. Here weak equivalence means that i induces a bijection on sets of components (0-cells modulo being joined by a 1-cell) and, for every 0-cell x of X and every n, a bijection on homotopy groups, which are classes of n-loops based at the degenerate cells above x, identified when joined by an (n+1)-cell.
The selected goal is the elementary-expansion theorem. It does not itself assert the complete equivalence between the homotopy theory of spaces and all coherator models. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The conclusion requires preservation of components and every based homotopy group, for cellular models built through transfinite attachments.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Tensorial absorption is a structural regularity property of a C*-algebra. A criterion in the central-sequence algebra would detect it through asymptotically commuting elements. The pinned manuscript supplies the research context.
For a nontrivial separable unital C*-algebra A and a free ultrafilter, the central algebra is the commutant of A's diagonal copy inside its norm ultrapower.
The selected goal is OAI.KirchbergRordam.character_criterion. Its central assertion is
The theorem states that, for a nontrivial separable C*-algebra A (in the base universe Type) and a free ultrafilter ω on the natural numbers, meaning an ultrafilter that refines the cofinite filter, the norm central-sequence algebra of A has no characters if and only if A is isomorphic to its minimal (spatial) C*-tensor product with the Jiang–Su algebra. The norm ultrapower of A is the quotient of the C*-algebra of bounded sequences in A by the ideal of sequences whose norms tend to zero along ω; the central algebra is the closed star-subalgebra of this ultrapower consisting of elements commuting with the diagonal copy of A, the images of constant sequences. The central algebra has no characters means that every star-homomorphism from it to ℂ (a linear, multiplicative, star-preserving map, not required to preserve the unit) is the zero map. The Jiang–Su algebra is the inductive limit C*-algebra of the standard prime-dimension-drop model system. The right-hand side asserts the existence of a star-algebra isomorphism over ℂ from A onto this tensor product. The proof is admitted in the source.
The target is the character criterion for one algebra and one free ultrafilter. The manuscript's infinite tensor-power consequence is not separately attached. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The two directions relate a scalar-valued representation obstruction to an isomorphism with a tensor product, using the precise inductive-limit Jiang–Su model.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Uniqueness of irreducible representations is a strong condition on a C*-algebra. The target asks for a counterexample to the expected compact-operator characterization without added set-theoretic hypotheses. The pinned manuscript supplies the research context.
Representations are nonzero star homomorphisms on complete complex inner product spaces. Irreducibility is expressed by closed invariant subspaces, and equivalence by an intertwining linear isometry.
The selected goal is OAI.NaimarkZFC.main. Its central assertion is
The theorem states that, for every universe level choice, there exists a type A carrying a C*-algebra structure (over ℂ) with the following five properties. First, A is simple in the sense that it is nontrivial and its only closed two-sided ideals are {0} and A itself. Second, A is not finite-dimensional as a complex vector space. Third, there is a continuous linear functional τ : A → ℂ that is a faithful tracial state: τ has norm 1, τ(aa) is a nonnegative real number for every a, τ(ab) = τ(ba) for all a and b, and τ(aa) > 0 whenever a ≠ 0. Fourth, A has a unique irreducible representation up to unitary equivalence: whenever π and ρ are nonzero star-homomorphisms (non-unital algebra homomorphisms preserving star) from A into the bounded operators on complete complex inner product spaces H and K, taken in the stated universe levels, and each has no closed invariant subspace other than 0 and the whole space, there is a linear isometric isomorphism U : H → K with U(π(a)x) = ρ(a)(Ux) for all a and x. Fifth, for every complete complex inner product space H in the stated universe, A is not isomorphic to the algebra of compact operators on H, meaning there is no injective star-homomorphism φ : A → B(H) whose range is exactly the set of compact operators on H.
The target explicitly requires simplicity, infinite dimension, a faithful tracial state and failure of all compact-operator isomorphisms in the stated universes. No additional set-theoretic hypothesis appears in its binders. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The construction must combine infinite dimension, a faithful trace and uniqueness across the specified representation universes.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Quasitraces behave linearly on commutative subalgebras but need not initially be additive on all positive elements. A uniform defect on a fixed pair would separate them from traces. The pinned manuscript supplies the research context.
The algebra is separable and carries a normalized 2-quasitrace, meaning a quasitrace with the specified extension to two by two matrices. The witnesses are positive contractions.
The selected goal is OAI.Kaplansky.kaplansky_quasitrace_counterexample. Its central assertion is
The theorem (admitted in the source, not proved there) states that there exists a separable C*-algebra A, with its complex C*-algebra structure, for which the following hold. First, A carries a normalized 2-quasitrace. Here a 1-quasitrace is a function τ : A → ℂ such that τ(xx) is a nonnegative complex number for all x; τ(xx) = τ(xx*); τ(h + i k) = τ(h) + i τ(k) for self-adjoint h and k; and the restriction of τ to every norm-closed commutative (non-unital) star-subalgebra over ℂ is ℂ-linear. It is normalized if τ(1) = 1, and it is a 2-quasitrace if it extends to a 1-quasitrace σ on the 2×2 matrices over A (as a C*-matrix algebra) satisfying σ of the matrix with a in the upper-left corner and zeros elsewhere equal to τ(a) for all a. Second, there are positive contractions a and b in A, meaning each is of the form x*x and has norm at most 1, such that every normalized 2-quasitrace τ on A satisfies Re(τ(a+b) − τ(a) − τ(b)) ≥ 1/144. So no such τ is additive on this pair.
The selected goal is the quantitative quasitrace counterexample. The stable-finiteness and tensor-product consequences are retained as a separate published bundle. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The same positive contractions must exhibit the defect for every normalized 2-quasitrace, while at least one such quasitrace exists.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
adc7f1241b42. Selected goal source.Norm ultrapowers permit approximations along a free ultrafilter. An explicit algebra excluded from every nuclear target ultrapower tests the limits of this embedding mechanism. The pinned manuscript supplies the research context.
The group is the stated dyadic semidirect product. Its full group C*-algebra is specified by a universal property, and the ultrapower is bounded sequences modulo norm-null sequences along the ultrafilter.
The selected goal is OAI.NuclearUltrapower.main_no_embedding. Its central assertion is
The theorem states that, working in universe level 0, there exist a nonzero unital complex C*-algebra A and a group homomorphism ι from G into the unitary group of A such that the following hold. Here G is the semidirect product of the additive group of 3-vectors over the dyadic rationals Z[1/2] by SL₃(ℤ) × ℤ, where a matrix acts linearly on the vectors and the integer n acts by multiplication by 2ⁿ. First, (A, ι) is the full group C*-algebra of G: the span of ι(G) is dense in A, and every homomorphism of G into the unitaries of a nonzero unital C*-algebra D extends uniquely to a unital -homomorphism A → D. Second, A is separable as a topological space. Third, for every nonzero unital C-algebra B that is nuclear in the tensor-norm sense (minimal and maximal C*-tensor norms agree on B ⊗ C for every C*-algebra C) and every free ultrafilter ω on ℕ (one containing no finite set), there is no injective unital -homomorphism from A into the norm ultrapower of B, namely bounded B-valued sequences modulo those tending to 0 along ω. Fourth, there exist a nonzero unital C-algebra O and two elements s₀, s₁ of O satisfying the Cuntz relations (each sᵢsᵢ = 1 and s₀s₀ + s₁s₁* = 1), with O universal for these relations, such that for every free ultrafilter ω there is likewise no injective unital *-homomorphism from A into the norm ultrapower of O. The theorem is admitted in the source, not proved.
The goal includes separability, the full group universal property, nonembedding and the Cuntz-algebra corollary in the stated base universe. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
A single source algebra must obstruct all nuclear targets and free ultrafilters. Universal full-group and Cuntz-algebra constructions are part of the target.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Hilbert-space spectral localization does not automatically produce bounded elements of the underlying operator algebra. Bounded recovery asks when positive averaged detection survives that passage. The pinned manuscript supplies the research context.
A standard modular datum has a cyclic separating vector, spectral calculus and scalar centralizer. Unit vectors lie in shrinking bands around a fixed real spectral parameter.
The selected goal is OAI.BoundedRecovery.bounded_recovery. Its central assertion is
The theorem states that, for a standard modular datum S on a complex Hilbert space H (a von Neumann algebra M with a unit cyclic and separating vector ξ, a nondegenerate real spectral calculus D with unitary group D.unitary t, an antilinear isometric involution J, and a closed Tomita graph of a ↦ (aξ, a*ξ) over a ∈ M equal to the pairs (p, Jq) where p is mapped to q by the half-exponential graph of D) satisfying ScalarCentralizer (every a in M fixed by conjugation with D.unitary t for all real t is a complex multiple of the identity), the following holds. Let ω be an ultrafilter on ℕ that refines the at-infinity filter (ω ≤ atTop), let T be a bounded complex-linear operator on H, let s be real, and let δₙ > 0 tend to 0. Suppose hₙ are unit vectors, each with spectral support in the interval [s − δₙ, s + δₙ] (every bounded continuous function vanishing on that interval annihilates hₙ under the calculus), and suppose the limsup over n of the ω-limit of the symmetric time averages (1/2k)∫{−k}^{k} ‖T(D.unitary t hₙ)‖² dt is strictly positive. Then the recovery conclusion holds: there are a strictly increasing sequence nⱼ, operators vⱼ in M, and constants C and η > 0 such that for every j, ‖vⱼ‖ ≤ C, ‖T(vⱼ ξ)‖ ≥ η, and vⱼξ has spectral support in [s − 4δ{nⱼ}, s + 4δ_{nⱼ}]. The proof is admitted, not supplied.
The hypothesis is positivity of the specified ultrafilter time average and outer limsup. Bicentralizer triviality and further rigidity applications are not separate conclusions of this target. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The recovered vectors must come from uniformly bounded algebra elements while retaining detection and shrinking spectral support along a subsequence.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.A bounded algebra representation need not visibly preserve adjoints. The similarity problem asks whether one bounded change of Hilbert-space coordinates restores that structure. The pinned manuscript supplies the research context.
Representations are continuous unital complex algebra homomorphisms into bounded operators on a complete complex inner product space. The conjugating operator must have a bounded inverse.
The selected goal is OAI.KadisonSimilarity.similarityTheorem. Its central assertion is
The theorem states that, for every C*-algebra A in universe u and every complex inner product space K in universe v that is complete (a Hilbert space), every bounded unital representation of A on K is similar to a star representation. Here a bounded unital representation is a continuous ℂ-algebra homomorphism π from A into the algebra of bounded linear operators on K; being an algebra homomorphism of unital algebras, it preserves the identity. Similarity to a star representation means that there is an invertible bounded operator S on K, with bounded inverse, such that for every a in A, S π(a*) S⁻¹ equals the adjoint of S π(a) S⁻¹. Equivalently, the conjugated map a ↦ S π(a) S⁻¹ commutes with the involution. This is the Kadison similarity statement as a defined proposition SimilarityTheorem, and the source declares it as a theorem whose proof is admitted rather than supplied.
Similarity is the selected goal. The two commutator formulations and explicit hyperreflexivity estimate remain distinct references with their own constants and universes. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
One operator S must work simultaneously for every algebra element. Uniform matrix-amplification estimates cannot acquire a dimension-dependent constant.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
OAI.KadisonSimilarity.universalCommutatorTheorem (Open).OAI.KadisonSimilarity.universalHyperreflexivity (Open).OAI.Kadison.uniform_commutator (Open).adc7f1241b42. Selected goal source.A hierarchy of infinitely many overlap scales need not fill an interval. Full support asks whether an admissible zero-temperature SK minimizer leaves any gap below overlap one. The pinned manuscript supplies the research context.
An order parameter is a nonnegative, monotone, right-continuous integrable function on [0,1). The Parisi value is defined by a Brownian stochastic-control supremum. The Stieltjes measure μ lives on the subtype Time=[0,1); its support below is taken in that relative topology.
The selected goal is OAI.ZeroTemperatureSK.full_support. Its central assertion is
The theorem states that, for any probability space carrying a real Brownian motion B (a BrownianSystem W) and any order parameter γ on [0,1), if γ minimizes the Parisi functional over all order parameters, then a five-part full-support conclusion holds. An order parameter γ:[0,1)→ℝ is nonnegative, monotone, right-continuous and integrable (extended by zero outside [0,1)). For a time t and position x, the value is the supremum, over controls α progressive for the filtration generated by the Brownian increments after t and bounded by 1 in absolute value, of the expected payoff |x + B₁ − B_t + ∫_t^1 γ(s)α(s−t)ds| − ½∫_t^1 γ(s)α(s−t)²ds. The gradient and curvature are its first and second derivatives in x, and the Parisi functional is value at (0,0) minus ½∫_0^1 tγ(t)dt. Being a minimizer means the functional at γ is at most its value at every order parameter η. The conclusion says: (1) there is a measure μ on [0,1) with μ((−∞,t]) = γ(t) for every t in [0,1); (2) there is a diffusion X, a process progressive for the Brownian filtration that almost surely is continuous on [0,1], starts at 0, and satisfies X_t = B_t + ∫_0^t γ(s)·gradient(s,X_s)ds; (3) every such measure μ has full support, equal to all of [0,1); (4) γ is strictly increasing, γ(a)<γ(b) whenever a<b; and (5) for every such diffusion X and every t in [0,1), the expected square of the gradient at X_t equals t, and the expected square of the curvature at X_t equals 1.
The main target is the five-part full-support conclusion for every encoded minimizer and Brownian system. The separate value-consequences bundle is retained as an additional formal target. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The conclusion couples support, strict increase, diffusion existence and derivative identities. Minimization alone is not an assumption of those conclusions.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
Additional published targets are included as separate references:
OAI.SKValue.value_consequences (Open).adc7f1241b42. Selected goal source.Vertex operator algebras describe conformal theories algebraically, while conformal nets describe local operator algebras. Passing between them requires analytic control of fields. The pinned manuscript supplies the research context.
The selected algebra is simple, unitary, CFT-type and strongly rational. Its modes act on an inner product space, and smeared fields act on the Hilbert completion.
The selected goal is OAI.MinimalVertex.main. Its central assertion is
The theorem states that, for a CFT-type vertex operator algebra A on a complex inner product space V (a vertex algebra with vacuum, state-field map and Jacobi identity, together with a conformal vector, central charge, finite-dimensional graded pieces with degree-zero part spanned by the vacuum, conformal vector in degree 2 acting as the grading operator, and the Virasoro relations), equipped with a unitary structure U (an antilinear involution fixing the vacuum and conformal vector, compatible with all modes, together with a unit-norm vacuum and an invariance relation between a mode and the corresponding mode of the transformed adjoint-side vector), if A is simple (nonzero vacuum and no ideals other than 0 and the whole space) and strongly rational (self-contragredient, rational in the sense that every admissible weak module is completely reducible, and C2-cofinite), then three things hold. First, A has polynomial energy bounds: for every a in V there are C>0 and natural numbers p,k with ||a_(n) b|| ≤ C(1+|n|)^p ||(1+L_0)^k b|| for all integers n and all b in V. Second, the CKLW strong locality property holds: all vectors satisfy these bounds, and for every proper circle arc I the von Neumann algebra generated by closed smeared fields supported in I, built on the completion of V, lies in the commutant of the algebra attached to the complementary arc. Third, the assignment of these interval algebras to proper arcs admits the structure of an irreducible conformal net, meaning a separable Hilbert space with isotony, locality, a continuous Mobius representation extended to a continuous projective representation of smooth circle diffeomorphisms with covariance and locality of the action, a unit invariant vacuum that is unique up to scalar and cyclic, a positive self-adjoint Hamiltonian generating the rotation flow, and trivial commutant of all interval algebras apart from scalars.
The target proves polynomial energy bounds, the specified CKLW strong locality property, and existence of an irreducible conformal-net structure. Complete rationality, category equivalences and extension classification are not conclusions of this reference. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
Formal algebraic identities must yield polynomial operator bounds and locality for closed smeared fields on the completion.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.Bounded-error factoring and exact factoring have different correctness requirements. Repetition until success and increasingly accurate rotations do not automatically give a fixed-gate, worst-case polynomial exact algorithm. The pinned manuscript supplies the research context.
Circuit instructions use NOT, CNOT, Toffoli, Hadamard and phase primitives, their inverses and singly controlled versions. A classical finite-stack machine generates each encoded circuit from unary input length.
The selected goal is OAI.ExactQuantumFactoring.exact_quantum_factoring. Its central assertion is
The theorem states that the defined proposition MainTheorem holds, i.e. there is a family of quantum circuits indexed by input bit length ℓ with three properties. Circuits are lists of instructions on q qubits, each applying one of 20 named gates (the primitives NOT, CNOT, Toffoli, Hadamard and phase, each optionally inverted and optionally given one extra control) to distinct wires; the output state is obtained by applying the instructions in order to the basis state holding the binary digits of N, least significant bit first, on the first ℓ wires, with all other wires zero. First, the family is uniform: a single Turing machine (TM2) with finite stack alphabets computes in polynomial time the encoding of the ℓth circuit from the unary string of length ℓ, where the encoding writes the qubit count, instruction count and each instruction's gate code, inverse and control flags and wire indices in unary. Second, the numbers of qubits and of instructions of the ℓth circuit are both bounded by one fixed polynomial in ℓ with natural-number coefficients. Third, for every ℓ and every integer N ≥ 2 whose binary length is exactly ℓ, the circuit has at least ℓ + n² qubits, where n = max(128, ℓ), and the total squared amplitude on basis states whose output is correct equals exactly 1. A basis state is correct if reading n consecutive blocks of n bits after the input wires, each block as a binary number, gives the nondecreasing list of prime factors of N with multiplicity, padded with zeros to length n.
The output is a sorted list of prime factors with multiplicity in fixed binary blocks, padded with zeros. The target is an existence theorem for a uniform circuit family, not an uploaded executable factoring program. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
Uniform generation, polynomial qubit and gate bounds, a fixed finite gate alphabet, and probability-one correctness must all hold together.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.The parity function tests whether a shallow quantum circuit can combine information from all input bits. Polynomially many ancillary qubits make the measured-output formulation substantially stronger than a restriction to clean final registers. The pinned manuscript supplies the research context.
Allowed gates are arbitrary one-qubit unitaries and Toffoli gates with finitely many controls. Gates within a layer have disjoint supports. Inputs occupy the first n qubits, with zero ancillas.
The selected goal is OAI.QAC.parity_lower_bound_polynomial_size. Its central assertion is
The theorem states that, for any positive integers d, k and C, there is a threshold n₀ such that for every n ≥ n₀ and every total qubit count N with n ≤ N ≤ C(n+1)^k, the following holds. Take any circuit given as a list of at most d physical layers on N qubits, where a layer is a list of gates with pairwise disjoint supports and each gate is either a one-qubit unitary on a single qubit or a Toffoli gate with an arbitrary finite set of control qubits and a target outside that set (flipping the target exactly when all controls are 1). The circuit's matrix is the product of the layer matrices, with the first layer acting first. For every choice of output qubit out among the N qubits, there exists an n-bit input x such that successProbability, the total Born probability over all final computational-basis strings y whose out-th bit equals the parity (sum mod 2) of x, is strictly less than 2/3. Here the input state is x placed on the first n qubits with all remaining qubits set to 0, and the remaining qubits are summed over with no requirement that they be clean. So fixed-depth circuits with polynomially many qubits cannot compute parity with worst-case success probability at least 2/3. The theorem is admitted in the source (proof is sorry).
The published goal fixes success threshold 2/3 and positive integer depth, polynomial exponent and coefficient. It does not state the manuscript's full range of positive advantages. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.
The bound covers arbitrary gates, all choices of output qubit and polynomially many total qubits. Discarded registers can retain unrestricted garbage.
The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.
The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.
adc7f1241b42. Selected goal source.