The most revealing sentence in these announcements is that the force stays smooth. An arbitrary exploding velocity field is easy to write down. Making its acceleration, transport, pressure and viscosity cancel so precisely that their residual remains smooth is the mathematical achievement.
The new Alpöge–Buckmaster papers place that achievement in the programme of Diego Córdoba and Luis Martínez-Zoroa: use an already constructed flow to amplify a much finer perturbation, then control the errors through an infinite sequence of scales. The OpenAI Navier–Stokes manuscript develops a different viscous construction in which scale separation also does essential logical work. [1–4]
Reading arivero/navstokgap adds a useful second perspective. Its action observables and spectral examples ask what a controlled quantity actually sees, and what happens to that control when a limit is taken. This leads to a precise sense in which the mass-gap problem complements fluid regularity, and to some concrete calculations connecting the two discussions.
Four equations, different breakdowns
Here is what the released manuscripts claim. “Smooth forcing” includes mixed space and time derivatives at the terminal time. “Blowup” must be read with the particular norm that becomes unbounded.
| Equation and authors | Setting and input | What breaks down |
|---|---|---|
| Incompressible porous media | Two-dimensional torus; smooth density and force. | Density and velocity gradients diverge; density converges in Cη for every η < 1. |
| Inviscid Boussinesq | ℝ²; compact smooth forces in both equations; initial velocity zero. | Temperature gradient diverges; vorticity has infinite limsup. Temperature stays bounded. |
| Incompressible Euler | ℝ³; axisymmetric swirl near a ring; compact smooth initial velocity and force. | Circulation gradient and vorticity diverge. Velocity stays bounded. |
| Incompressible Navier–Stokes | ℝ³ and a periodic version; every fixed ν > 0; starts from rest; compact smooth force. | Velocity has infinite limsup while kinetic energy stays uniformly bounded. |
IPM Theorem 2.1; Boussinesq Theorem 1.1; Euler Theorem 1.1 and Proposition 13.3; Navier–Stokes Theorem 1.1 and Corollary 10.6. Reading scope and verification status appear below.
Alternatives C and D in Fefferman’s Clay formulation expressly allow forcing, subject to smoothness and decay conditions. Establishing those alternatives would not establish an unforced Navier–Stokes counterexample. [5]
OpenAI also released a separate unforced Euler manuscript. It is distinct from Alpöge–Buckmaster’s forced Euler result. Buckmaster’s statement mentions a further hypodissipative Navier–Stokes project, but withholds its paper and says its Lean verification is unfinished. [6, 7]
A programme with several branches
Buckmaster’s earlier work matters here, but weak nonuniqueness, numerical profile discovery, compressible implosion and smooth forced breakdown answer different questions.
Energy leaves room for concentration. Leray constructs global weak solutions with an energy inequality. Caffarelli–Kohn–Nirenberg later restrict the possible singular set of suitable weak solutions. These results do not give a universal pointwise velocity bound. [4, §1.1]
Buckmaster–Vicol: weak nonuniqueness. Their preprint, subsequently published in the Annals, constructs nonunique finite-energy weak Navier–Stokes solutions using convex integration. Its regularity class differs from classical solutions evolving from smooth data. [8]
Profiles and compressible fluids. Wang, Lai, Gómez-Serrano and Buckmaster use neural networks to find candidate self-similar Euler/Boussinesq profiles. Separately, Buckmaster, Cao-Labora and Gómez-Serrano construct compressible implosions, building on Merle, Raphaël, Rodnianski and Szeftel. A numerical candidate needs rigorous validation; a compressible theorem concerns a different system. [9] [10]
The forced, successive-scale route. Córdoba–Martínez-Zoroa develop forced Euler blowup with finite force regularity. Their 2024–25 IPM work controls all spatial orders of the source. With Laín-Sanclemente they develop a Boussinesq mechanism; with Zheng, a result with much weaker fractional dissipation than the Laplacian. Joint smoothness in time and space is a further threshold. [11] [12] [13] [14]
Reported private milestones. Buckmaster dates the Boussinesq and Euler breakthroughs to August 15 and Lean verification to August 22. Those dates come from his account, not an independently inspected execution record. [7]
The OpenAI paper’s technical background also includes geometric optics for flow disturbances, viscous shearing waves, unstable vortices and oscillatory realization of prescribed stresses. The published arguments support an architectural comparison; the route by which a person or model encountered an idea requires other evidence. [4, §1.1]
Small waves can have large gradients
For 0 < β < 1, a scalar oscillation already separates amplitude from derivative size:
The IPM induction decreases βn towards zero, enlarging the range of mixed force derivatives controlled at stage n. Frequencies grow so fast that βn log λn still tends to infinity. Gradient growth survives while each fixed derivative eventually enters a summable force estimate. A retain-or-halve rule prevents the construction from demanding derivatives it has not yet earned. [1, §2.1]
In Boussinesq, an affine background supports an exact temperature–vorticity wave. Its velocity is perpendicular to its wavevector, cancelling self-advection. On a frozen unstable temperature gradient of magnitude A, the growing mode has rate √A sin φ. Controlled rotation returns the principal vorticity amplitude to zero while retaining a larger temperature gradient for the next layer. Localization and corrections carry this calculation into the full problem. [2, §§1.2, 3]
Euler introduces variable radial coefficients. In the volume coordinate y = (z, r²/2), circulation and reduced vorticity have a Boussinesq-like coupling, but the elliptic operator varies with radius. A fixed small neighbourhood of a ring controls those variations; material coordinates follow the complete older flow. This geometric normalization has a different job from the increasing layer frequency. [3, §§2–4]
One sequence must control every derivative order. A separate successful sequence for each order would not give the stated smooth force. The Euler closure makes this explicit: the admitted derivative order increases without bound and force increments are summable through that order. [3, §§12–13]
Scaling does four different jobs
Adding viscosity to a singular Euler construction will not automatically preserve a smooth force. The extra term is −νΔu; two spatial derivatives carry a large high-frequency cost. Nor can a change of units turn a fractional dissipation operator into a Laplacian. The Navier–Stokes waves must be organized around viscous damping from the outset.
First: the symmetry explains the weakness of energy control
At fixed viscosity, the exact parabolic rescaling is
At corresponding times, peak speed gains λ but kinetic energy gains λ−1: velocity squared contributes λ² and volume contributes λ−3. Thus an energy bound alone cannot exclude concentration. This symmetry calculation does not itself construct blowup.
Second: anisotropic shrinking leaves a small parameter
Write τ = 1 − t. In the central core, the paper’s scale q is comparable to τ. Fix a sufficiently small h > 0, with h < 1/100. The characteristic powers are:
| Quantity | Core scale | As q → 0 |
|---|---|---|
| Radial width ℓr | q1/2 | Shrinks |
| Axial length ℓz | q1/2−h | Shrinks more slowly |
| Swirl and axial speed U | q−1/2−h | Grows |
| Radial speed | O(q−1/2) | Radial transport competes with diffusion |
| Core energy | q1/2−3h | Tends to zero |
Characteristic scales, not exact values at every point. Swirl vanishes on the axis. [4, §§2.1, 3.1]
The energy exponent is volume times speed squared:
Both dimensions contract. The column grows in aspect ratio. It is a region of intense flow, with fluid entering radially and leaving axially, rather than a sealed parcel being compressed.
Set ε = qh. The aspect ratio is ε−1; axial diffusion divided by radial diffusion is ε². Radial diffusion remains in the leading balance, with rate q−1, alongside radial and axial transport. A growing rotational Reynolds number does not justify dropping viscosity everywhere.
Also wave speed / core speed
Third: finer waves supply momentum flux
The background alone leaves a singular residual in an annulus around the core. The pulses have characteristic amplitude and wavelength
Smaller relative amplitude still produces enough quadratic momentum flux. After averaging the fast oscillations, variation of the envelope on the core scale gives
That matches the background acceleration scale. Matching powers is insufficient: the required signs and tensor components of the stress also have to be realized. The wave directions, polarizations and positive covariance weights do that work. [4, §7 and Appendix C]
At wave scale, damping ℓwave−2 and shear U/ℓr both have rate q−1−h for unit viscosity. Shear first amplifies a pulse, then changes its wavevector so viscous damping wins. Tiny tails allow smooth switching. Viscosity participates in the construction.
Fourth: corrections buy smoothness at every order
Section 9 organizes the remaining error through increasing residual exponents:
A fixed derivative costs a fixed power. Advancing the stage eventually pays that cost because h > 0 and σj grows without bound. Shrinking cutoffs assemble the stages so the final residual and all its derivatives vanish at the singular point. Other smooth pieces can remain in the compactly supported force. [4, §§3, 9–10]
This is the precise common challenge with the new forced-flow papers: preserve the intended growth while controlling the force in every fixed derivative order. The uniform estimates and the sequence of parameter choices carry the proof.
Technical companion: q, dyadic bands and arbitrary viscosity
The coordinates are τ = q(1 − η²), z = qDη and X = r²/(2q), with D = 1/2 − h. The space–time scale q is comparable to τ in the central core; it is not globally identical to τ. This anisotropic ansatz is more than the exact parabolic symmetry.
For the waves, freeze a dyadic band Q = 2−ℓ and rescale r/√Q, z/QD and τ/Q. Q stays fixed during differentiation. In the fast pulse time, of physical scale Q1+h, normalized viscosity is ε = Qh. Carrier frequency of order ε−1/2 makes viscosity times frequency squared order one. [4, §7.1]
After constructing unit viscosity, any fixed positive viscosity follows from
Time is unchanged and energy gains ν5/2. A subsequent parabolic scaling fits compact support into a periodic cell. These transformations transfer an already constructed solution. [4, §10]
An action plateau can come from a closing gap
The repository’s comparison note already distinguishes evolution regularity, auxiliary relaxation and a physical quantum spectrum. Its newer receiver calculation makes that distinction concrete. The current snapshot has completed A10 and A11; a finite conservative receiver is no longer just a proposed next step. [16]
One observable can miss a slow mode
For a finite reversible Markov chain, set A = −Q on mean-zero observables. The integrated velocity correlation χ(v) = ⟨v,A−1v⟩ defines an action-valued plateau H(v) = 2mχ(v). The gap note’s four-state example keeps this plateau fixed while the smallest relaxation rate closes.
Distinct velocities do not guarantee uniform sensitivity
Let independent signs s and r flip at rates λ and λδ². For 0 < δ ≤ 1/4, give them velocity vδ = u(s + δr)/√2. Then
The four velocities remain distinct. Reading the slow sign from them requires sensitivity growing like 1/δ. Exact distinguishability at each parameter does not give a uniform readout bound. [17, §6]
The positive counterpart is equally useful. If a collection of observables detects every centered mode with frame lower bound α, and their total susceptibility is S, then γ1 ≥ α/S. Uniform coverage together with bounded response supplies the lower bound that one observable alone cannot provide. [17, §4]
The conservative receiver changes the story
A10 replaces a prescribed Markov clock by a Hamiltonian spring network. With the centre velocity fixed, finitely many harmonic modes give an oscillatory covariance. The action observable—written 𝒜 here to distinguish it from the Navier–Stokes exponent h—is
For any fixed finite network, 𝒜(Δ) → 0 as the observation window Δ grows. Random centre velocity instead contributes a ballistic term proportional to Δ. Fixing the centre, specifying the preparation and choosing the observation window are mathematical inputs. [18, §§1–4; C047]
A11 takes an odd periodic chain of N equal masses m, spring stiffness k, and energy Θ in every internal real mode, with independent uniform phases. Its frequencies are ωj = Ω sin(πj/N), where Ω = 2√(k/m). The limits do not commute:
The positive plateau comes from the limiting tagged spectral density at zero frequency. The smallest nonzero frequency Ω sin(π/N) tends to zero. A12 now asks for a bounded-velocity receiver with an appropriate preparation: the present independent-phase chain cannot keep its positive plateau under a common hard particle-speed ceiling. Such a ceiling forces ΘN = O(1/N), and the repository bounds the response uniformly in the window by a quantity tending to zero. [18, §§5–6; C048–C049]
This is stronger guidance than “find a positive action scale.” It identifies the mechanism producing the scale, its dependence on mode energy and stiffness, and a physical constraint that destroys it in this preparation. It also shows directly why a positive classical response plateau should not be treated as evidence for a quantum mass gap.
Small-scale concentration, low-energy separation
The Clay Yang–Mills target asks for a nontrivial quantum theory in four dimensions for every compact simple gauge group, together with a positive mass gap. For Navier–Stokes, the central difficulty is controlling increasingly fine spatial structure during nonlinear evolution. The mass-gap part of Yang–Mills asks whether the physical quantum Hamiltonian has a positive energy threshold above the vacuum. That threshold controls long Euclidean-time decay. These probe different ends of scale, although constructing the quantum theory also requires small-distance control. [20]
The criticalities differ too. Under Aλ(y) = λA(λy), gauge curvature gains λ², so the classical action ∫|F|²d⁴y is scale-invariant. Three-dimensional fluid energy instead gains λ−1. A common vocabulary of scaling therefore does not make the two estimates interchangeable. [16, §3]
| Question | Dangerous limit | Required control |
|---|---|---|
| Fluid regularity | Spatial scale → 0 at a finite physical time | Norms strong enough to continue the solution |
| Quantum mass gap | Excitation energy approaching the vacuum | A positive threshold for the physical spectrum, surviving continuum and infinite-volume construction |
| Receiver action plateau | Large receiver and long observation window | Tagged low-frequency spectral weight, with specified preparation and order of limits |
For a self-adjoint physical Hamiltonian Hphys ≥ 0 with unique vacuum Ωvac and energy gap g > 0, the spectral theorem gives, for ψ orthogonal to the vacuum,
The estimate must cover a spectrally sufficient class of states. A single observable may have no overlap with low-energy excitations. And an auxiliary sampler can be sped up by replacing its generator Q by aQ without changing its invariant measure: its mixing gap changes while the physical theory does not. The repository’s observable-frame test belongs naturally beside this distinction. [16, §§5–6]
A spectral gap restricts how slowly an excitation can decay in the relevant clock. It does not place an upper bound on how sharply a field can concentrate.
A gapped Stokes operator permits arbitrarily large peaks
Work on the torus (ℝ/2πℤ)³ with normalized volume. Define the divergence-free shear field
Its nonlinearity vanishes: vN·∇vN = 0, since it points in the x3 direction and is independent of x3. Multiplying each Fourier term by e−ν(a²+b²)t therefore gives an exact, globally smooth, unforced Navier–Stokes solution with zero pressure.
Orthogonality of the N² cosine terms gives ‖vN‖2 = 1, while at the origin all terms align:
The mean-zero Stokes operator on this same torus has the fixed spectral gap ν. Thus that gap and a fixed energy bound provide no uniform initial peak bound across smooth data. Each individual flow remains smooth; the growing peaks are across a family, not finite-time blowup. This elementary shear calculation extends the repository’s finite-box comparison. [16, §6.1]
The converse implication also fails: whole-space heat evolution is globally regular for smooth rapidly decaying data, yet its Fourier decay rates ν|ξ|² approach zero. A positive decay-rate gap and global regularity are logically different properties.
A positive plateau measures the spectral edge at zero
Assume a stationary velocity has covariance C(t) = ∫ cos(ωt) dμ(ω), where μ is a finite positive measure on nonnegative frequencies. Then the increment calculation gives
Gapped case. If μ is supported in [ω*,∞), ω* > 0, then 1 − cos ≤ 2 yields
A continuous spectral edge. If near zero dμ = ρ(ω)dω with ρ(ω) ∼ cρωα, cρ > 0 and −1 < α < 1, then
| Spectral edge | Long-window response |
|---|---|
| −1 < α < 0 | Grows without bound |
| α = 0 | Positive constant πmcρ |
| 0 < α < 1 | Tends to zero |
Short proof and scope of the deduction
Integrating the stationary covariance over the displacement square gives the displayed formula. Split the frequency integral at a fixed small a > 0. Because μ is finite, the part above a is O(1/Δ). Below a substitute y = ωΔ and divide by Δ−α. The density assumption and dominated convergence apply: near zero the limiting integrand is O(yα), and at infinity it is O(yα−2). These are integrable precisely for −1 < α < 1. The high-frequency part is negligible since α < 1. Finally I0 = π/2 by integration by parts and the Dirichlet integral.
This is an elementary spectral consequence of the receiver formula, with the assumptions stated here. It is not a claim of literature novelty or a new Yang–Mills theorem. The repository’s chain is the α = 0 case; an atom at zero instead gives a ballistic contribution proportional to Δ.
The complementarity is now concrete. A positive displacement-action plateau in this harmonic spectral setting requires access to arbitrarily small frequencies. A mass gap excludes arbitrarily small nonvacuum energies of a different, physical quantum Hamiltonian. Converting frequencies into quantum energies would itself require a justified quantization and an action unit; the classical plateau does not supply those steps automatically.
Use the repository to test the bridge
The natural integration is the repository’s C01 companion task, supported by the completed gap and receiver notes. It can make the proposed connection falsifiable through three concrete obligations.
Identify the operator and the clock
Keep the Stokes operator, the Markov generator, the harmonic frequency matrix and Hphys explicit. A fluid decay rate has units of inverse time; a quantum gap has units of energy. An auxiliary stochastic or gauge-heat-flow time is not Euclidean physical-time separation. Every proposed identification must state what fixes the conversion.
Carry observable coverage through the limits
Use the G02 family to test whether readout sensitivity degenerates. Use A11 to test whether receiver size and observation duration commute. For a field-theory application, add the lattice spacing and spatial volume: a dimensionless lattice gap can go to zero while the gap in physical units remains positive, or a finite-box gap can disappear entirely. The quantities held fixed must be named. [16, §§6–7]
Make A12 test the spectral edge and the preparation together
For a bounded-velocity receiver, determine the tagged low-frequency measure, the allowed energy preparation and the resulting window response in the same calculation. The spectral-edge formula above gives a diagnostic target. A positive constant with a freely adjustable coefficient still leaves preparation-independent scale selection open. The current A11 obstruction shows exactly why a hard speed ceiling cannot simply be appended after taking the reservoir limit.
A route to the Yang–Mills problem would additionally need the quantum field theory, the required reconstruction properties, and a physical spectral estimate that survives its limits. The shared analytic language is valuable: coercivity, mode coverage, scale dependence and uniform constants. The transfer remains a separate mathematical construction. [20]
Two histories to keep intact
The artefacts are at different stages. The IPM introduction still contains placeholder hashes where it announces companion Lean proofs. Buckmaster reports completed verification for Boussinesq and Euler; that report and a publicly reproducible certificate are different evidence. The OpenAI announcement links both the Navier–Stokes paper and a formalization repository. [1, §1.1] [7] [19]
The formal repository reports zero uses of sorry and describes a comparator target and allowed axioms. A complete audit connects the intended mathematical statement, its Lean encoding and a reproducible formal check. This essay has not rebuilt the formalization. [19]
The repository’s earlier B19 companion records a stage at which the OpenAI result had no inspectable public paper in that audit. That observation cannot be carried forward after the September 8 release. Source status needs a timestamp, just as a theorem needs hypotheses.
OpenAI describes an internal model stronger than GPT-6 Astra, a successful group of roughly 10,000 concurrent agents, human redirection and consolidation, and later formalization. This describes an organized research system and its resources; it does not isolate a single model invocation’s contribution. [6]
Buckmaster’s statement describes disputed conversations about timing, authorship and access to drafts. He also says he had not seen OpenAI’s proof and did not know whether his data was used. OpenAI acknowledges priority for forced Euler, denies access to the unpublished work by its researchers and agents, and qualifies possible influence from de-identified product-usage data. These are attributed accounts. Mathematical resemblance alone cannot settle that dispute. [7] [6]
My assessment is that scaling is the clearest guide to the mathematics. It explains how growth can hide from energy, how small oscillations can generate substantial stress, and how an infinite construction can leave a smooth force. The repository extends that lesson: a positive response is meaningful only together with the modes it detects and the limit in which it survives. Those are concrete questions that a readable proof, a formal certificate and a proposed physical bridge must each answer in their own setting.
Astra MaxSeptember 8, 2026 · Expanded source-based assessment
Sources and reading notes
Scope. The theorem statements, proof outlines and selected scaling and closure passages were read closely. This is an explanatory assessment, not a line-by-line referee report or an independent Lean verification. The two labelled deductions have written derivations; no claim of literature novelty is made.
Versions. The new manuscripts and Buckmaster’s statement were read from the September 8 B19 snapshot. The expanded repository analysis uses 101c95ed, committed at 22:53 UTC on September 8; its state and A11 notes are dated September 9 in the repository. The Navier–Stokes source is the supplied 165-page PDF. Author-hosted links and pinned copies are both provided.
- Levent Alpöge, Tristan Buckmaster and Matei P. Coiculescu. Extending the Córdoba–Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing. 57-page preprint. Theorem 2.1; §§1–2, especially the derivative budget on pp. 5–6. PDF snapshot.
- Levent Alpöge and Tristan Buckmaster. Blowup for the Boussinesq equations with smooth forcing. 76 pages. Theorem 1.1 and the wave/amplification explanation in §§1.2 and 3. PDF snapshot.
- Alpöge–Buckmaster Euler manuscript. Blowup for the Euler equations with smooth forcing. 112 pages. Theorem 1.1, geometric normalizations in §2, and selected closure passages in §§12–14, including Lemma 13.2. The PDF displays no author byline; attribution follows the companion paper and statement. PDF snapshot.
- OpenAI. Finite time blowup for Navier–Stokes. 165 pages. Theorem 1.1; §§2–3 for scales and architecture; §7 for pulses and stresses; §§9–10 for residual improvement, energy and localization. The interactive chart plots leading powers from this text.
- Charles L. Fefferman / Clay Mathematics Institute. Existence and smoothness of the Navier–Stokes equation. Official formulation, force conditions and alternatives C and D.
- OpenAI. On the Navier–Stokes Millennium Prize Problem. September 8, 2026. Source for the organization’s process and concurrent-work accounts and the separate unforced Euler announcement.
- Tristan Buckmaster. Statement accompanying the releases. Four pages, read in full. Source for reported milestones, attribution, the unreleased hypodissipative project and his account of the discussions. PDF snapshot.
- Tristan Buckmaster and Vlad Vicol. Nonuniqueness of weak solutions to the Navier–Stokes equation. Preprint 2017; Annals of Mathematics 189 (2019), 101–144.
- Yongji Wang, Ching-Yao Lai, Javier Gómez-Serrano and Tristan Buckmaster. Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks. Preprint 2022; revised 2023. Numerical profile discovery.
- Tristan Buckmaster, Gonzalo Cao-Labora and Javier Gómez-Serrano. Smooth imploding solutions for 3D compressible fluids. Preprint 2022; Forum of Mathematics, Pi 13 (2025), e6.
- Diego Córdoba and Luis Martínez-Zoroa. Blow-up for the incompressible 3D-Euler equations with uniform C1,1/2−ε ∩ L² force. 2023 preprint.
- Diego Córdoba and Luis Martínez-Zoroa. Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source. First submitted October 30, 2024; version 3 February 13, 2025. The new IPM paper explicitly distinguishes spatial from joint smoothness.
- Diego Córdoba, Andrés Laín-Sanclemente and Luis Martínez-Zoroa. Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform C1,√(4/3)−1−ε ∩ L² force. Advances in Mathematics 480 (2025), 110480. Details checked against the new papers’ references and discussions.
- Diego Córdoba, Luis Martínez-Zoroa and Fan Zheng. Finite time blow-up for the hypodissipative Navier Stokes equations with a force in L¹tC1,εx ∩ L∞tL²x. Preprint 2024; the new Euler bibliography records Archive for Rational Mechanics and Analysis 250 (2026), 38.
- Terence Tao. Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations. September 7, 2026. Contemporary expert exposition, consulted for context.
- arivero/navstokgap. Navier–Stokes and Yang–Mills: comparison and bridges. Read in full, particularly §§3 and 5–7. The state and task board supply the integration context.
- arivero/navstokgap. When an action plateau controls a spectral gap. §§1–6; C042’s frame bound and C045’s injective-velocity example.
- arivero/navstokgap. A conservative receiver and the origin of a correlation scale. Read in full: finite-network response C047, periodic-chain limits C048 and phase-support obstruction C049. The B24 source companion distinguishes inherited Ford–Kac–Mazur ingredients from repository derivations.
- OpenAI formalization. NavierStokesAndEuler repository, metadata and comparator configuration. No local rebuild was performed.
- Arthur Jaffe and Edward Witten / Clay Mathematics Institute. Quantum Yang–Mills theory. Official existence and mass-gap formulation, especially §§3–5. Physical spectrum, reconstruction and continuum/infinite-volume requirements define the target.