Collective geometry and the Yang–Mills gap

Jaffe and Witten state the problem for every compact simple gauge group 𝖦: prove that a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0, existence including axiomatic properties at least as strong as those of Wightman or of Osterwalder and Schrader. The theory’s data are a Hilbert space carrying a unitary representation of the Poincaré group of Minkowski space (ℝ⁴, η); positive energy, 0 ≤ H, which the Wightman axioms state as the joint spectrum of energy and momentum in the closed forward cone; a vacuum Ω, Poincaré-invariant and unique up to a phase, with HΩ = 0; and the gap, no spectrum of H in (0, Δ) for some Δ > 0, the supremum of such Δ, the mass of the theory, finite.

The metric η is fixed. It enters through Poincaré covariance and is not a variable of the theory: the energy of the states varies while the geometry does not respond to it. The problem is posed with the response of geometry to energy switched off.

The classical record is settled on the same background. Klainerman and Machedon proved global well-posedness of the classical Yang–Mills evolution on Minkowski space for finite-energy data, and Oh gave a second proof through the Yang–Mills heat flow. The measured record has no massless strongly interacting state, and lattice gauge theory computes a glueball spectrum with no state at zero mass.

A flat background holds only an empty source

FDT retains G_μν = κT_μν unchanged, with κ = 8πG/c⁴ and T_μν the one total source of the omnium, every channel entering once (I-⁠3, 0.5). Clocks and rulers are closed patterns adjusting their own dimensions to keep the chain, so their adjustment is the geometry and no background metric lies beneath them (0.1a). Every configuration whose geometry is η therefore has

Every state of the Clay theory off the vacuum carries energy. By the spectrum condition a state of zero energy is invariant under translations, so it is the vacuum, and every other state has E = ⟨ψ, Hψ⟩ > 0, gapped or not. That energy is a channel of the total gravitating energy, 𝓜 ≥ E/c², so R_s > 0 and α_ext(r_f) = R_s/r_f lies inside (0, 1) at every field radius: its geometry is not η. The vacuum has zero total source, ⟨Ω, T_μνΩ⟩ = 0: E = 0 and m = 0, two variables of the chain that 0.1 never lets vanish, and R_s = 0, no fold and no branch of the atlas. It is the empty state, and the omnium does not have one. Read as an operator, its stress does not vanish, T_μν(f)Ω ≠ 0 for every nonzero stress smeared over a bounded region, and the vacuum then fails the field equation on η with every other state.

No state of the Clay theory, as posed on its fixed background, is therefore a physical configuration. The statement holds the metric fixed while the energy of its states varies; it ignores general relativity by construction. If general relativity is ignored, the Clay problem statement does not apply to reality (Theorem 5).

Families with the metric held fixed

Along a differentiable family z(s) of complete solutions of I-⁠3 the field equation differentiates to DG[ġ] = κṪ + κ̇T. With the physical metric held fixed and the constants of the core fixed, the total source cannot move at first or at second order (Theorem 7):

This is the chain read as a constraint on families: when the energy of a pattern steps, the distances and durations it keeps step with it, and a family that changes the source while holding the metric fixed holds apart what the chain unifies. The constraint is on the total source, so channels compensate collectively at each point.

The pure-gauge interpolation is such a family. With θ = U⁻¹dU for a time-independent U, so that dθ + θ∧θ = 0, and A(s) = sθ on the fixed background in the temporal gauge, the field strength is F(s) = s(s − 1) θ∧θ and the Yang–Mills stress, quadratic in F at fixed metric and coupling, is

Its magnetic energy has a maximum at s = ½, the barrier a fixed-background analysis reads between two pure-gauge configurations. The family never enters the physical domain. Wherever θ∧θ ≠ 0 its source is nonzero at every s strictly between its ends, which the flat background does not hold; its two ends, where F = 0, are the empty state. At s = ½ it also fails the second-order condition that a family of complete solutions with the metric held fixed passes. The negative second derivative is a property of the interpolation on its background and says nothing about the stability of the omnium: the family is no deformation of it, and the stability of a pattern is the ledger form of I-⁠5b, never the sign of a second variation (0.1a).

The second variation and the gap criterion

The core names no objective for the omnium’s optimisation; the paper reads one, as its own reading, through an energy functional 𝓔 (0.1a). On the coupled domain, its second variation along an admissible curve through a constrained stationary point carries the response of geometry, source, support and phase together with the curvature of the constraint surface (Lemma 8):

A Hessian that varies a gauge connection with everything else held fixed is the second variation of a different functional along a different family. Free relaxation over responding coordinates only lowers the form, by the Schur complement Q₁₂Q₂₂⁻¹Q₁₂* ≥ 0 (Lemma 10): a positive frozen block decides nothing, and a negative one stays negative. A constraint adds the multiplier term −⟨λ, D²𝒞[u, u]⟩, whose sign is the curvature of the constraint surface, so the complete geometric response has no universal sign. What these forms bound is the spectrum of the generator of a closed pattern’s own region; the stability of a pattern is the ledger form of I-⁠5b.

The gap is then a uniform estimate on the generator of the internal dynamics of a closed pattern’s own region. With H_phys ≥ E_baseI, E_base > 0 its lowest eigenvalue, H_rel = H_phys − E_baseI and P₀ the ground projection,

In FDT the lower edge is a difference of confinement eigenvalues, ε_* = ħ(ω₁ − ω₀) = ħω₁ − E_base: the step from the lowest mode cost of a closed pattern’s own region, in which the one carrier stands, to the next (I-⁠2, I-⁠5), on the closure sector of I-⁠5a, which every transition keeps. Every mode cost is positive, so the gap is the step between the two lowest. The subtraction of E_base changes the reference for differences and introduces no empty state: E_base stays a channel of 𝓜. The confinement operator of the n = 3 closures, the class that carries colour, its domain and boundary data, and the estimate uniform over the admissible closures are to be derived (D-⁠1).

No empty region, and the infinite volume

An admissible region is a region of the physical spatial geometry that contains a ball of diameter d_min = 2ℓ_P; nothing below the floor is a region of anything (0.1a, 0.1b). Every admissible region holds omnium, whose energy is E = mc² with neither variable vanishing, so the energy measure has full support at admissible scales (Proposition 12). Full support alone gives no quantitative floor; the floor of the spacetime phase does, a positive infimum of its local density, to be derived from the omnium’s constitutive law (D-⁠3). The record exhibits it in the observed domain: the energy density of the substrate’s standing-wave state, u = aT⁴ = 4.17 × 10⁻¹⁴ J m⁻³ at 2.7255 K, fills that domain, its temperature the same in every direction to 1.2 × 10⁻³, the dipole, and to about 10⁻⁵ once the dipole is removed. A complete geometry of infinite volume is unbounded and holds infinitely many disjoint admissible regions, each carrying at least the floor e_*, so it carries at least Ne_* for every N:

On a geometry of infinite volume, the spatial ℝ³ of the Clay background, the energy measure of a configuration of the omnium is infinite, while the Clay vacuum assigns zero to every region. No subtraction of a finite constant carries the one to the other: the amount removed from each admissible region would be at least e_*, and its total over the volume is infinite. The floor’s own window grows with the region: a ball of radius R holding at least u has

which reaches the seam α_ext = ½ at R = 1.32 × 10²⁸ m. A fixed flat background holds no part of this energy. The gap statement is posed on a closed pattern’s own region, a bounded support within a universal domain (0.8), where E_base is finite and H_rel is defined.

The lowest pattern keeps time

Every pattern of the omnium has E_q = ħω_q > 0 and keeps the period T_q = h/E_q, longer than t_min = 2t_P: the thread it runs in that time is a physical stroke, and the floor bounds every stroke below (0.1a). A static pattern would have ω_q = 0, a vanishing variable of the chain. The lowest pattern of a closed pattern’s own region cycles with period T_base = h/E_base (Proposition 15), and in the spectral representation it stays in its eigenspace while its phase advances by 2π in that time:

That phase is the helical phase of I-⁠4. The Clay convention HΩ = 0 sets the period to infinity, a pattern that never completes a cycle, which is no pattern of the omnium. The energy accounting of an excitation therefore starts from a moving baseline: the excess is a difference of two complete configurations, each with its own geometry, network and every interaction counted once. Around every point the density is not uniform (0.1), so the lowest pattern carries the nodal architecture of P-⁠2, its density geodesics the channels along which its energy is transported. The self-consistent transport law of the lowest pattern and its network is to be derived (D-⁠2).

The paper settles the domain on which the gap is posed. Its commitments are the core’s, read through the problem, each with the result that ends it; its open items are named.

1. 0.1
   No massless strongly interacting state. The gap is the step between the two lowest confinement eigenvalues of a closed pattern’s own region (I-⁠2, I-⁠5), and the hadron spectrum has no massless strongly interacting state, as measured. A massless strongly interacting state ends this reading.
2. I-⁠3
   Fixed-background results, recovered. For a laboratory configuration the coupled equations reduce to the fixed-background ones with corrections of relative size α_ext, 2.65 × 10⁻³⁹ for 1 GeV in 1 fm: the classical well-posedness theorems are recovered, and the glueball spectra computed on the lattice are tests for the operator of D-⁠1, never its outputs. A Yang–Mills observable departing from its fixed-background value far above that size, with the source counted in full, ends the reading.
3. 0.8
   A floor in every region. Every admissible region carries energy, and in the observed domain the floor is exhibited by the substrate’s standing-wave state at u = aT⁴. A region of the observed domain found empty of energy ends the reading.
4. D-⁠1
   The gap between confinement eigenvalues. The operator D of the n = 3 closures, the class that carries colour, its domain and boundary data, and the estimate uniform over the admissible closures, with the observable sector reaching the lower edge, are to be derived (D-⁠1, with the core’s O-⁠9, and O-⁠6 for the gluon octet’s dynamics).
5. D-⁠2, D-⁠3
   The transport law and the floor’s value. The transport law of the lowest pattern with its density-geodesic network (D-⁠2) and the density floor of the spacetime phase with its value (D-⁠3) are to be derived.

For the same source and data FDT and general relativity solve the same equation and return the same curvature (I-⁠3). A laboratory Yang–Mills configuration of 1 GeV in 1 fm has 𝓜 = 1.78 × 10⁻²⁷ kg, R_s = 2.65 × 10⁻⁵⁴ m and α_ext(1 fm) = 2.65 × 10⁻³⁹. At that order the coupled equations reduce to the fixed-background equations, and the classical well-posedness theorems are recovered as the α_ext → 0⁺ reading of the coupled solution, with the geometric response a correction of relative size α_ext. The masses of closed patterns are the confinement eigenvalues of D-⁠1, whose operator is the core’s register item O-⁠9: the glueball spectra computed in lattice gauge theory are tests for it, never its outputs. The measured record, a hadron spectrum with no massless strongly interacting state, is accepted in full.

What FDT declines is the endpoint R_s = 0: the exactly flat background and the exactly empty vacuum of the problem statement, the endpoint of the one-sided limit α_ext → 0⁺, which no configuration attains (I-⁠6). The Clay statement poses pure Yang–Mills for every compact simple gauge group; in FDT the carrier stands only inside a closed pattern’s own region, and colour is the SU(3) of the internal nodal surface of the n = 3 closures (I-⁠2, I-⁠5): the physical gap problem is posed there. Where FDT, general relativity and quantum mechanics converge, FDT reproduces their results; where it declines their assumptions, here the fixed background and the empty vacuum, it is under no obligation to reproduce them.