The premises of Yang–Mills

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, a vacuum invariant and unique up to a phase with HΩ = 0, and the gap, no spectrum of H in (0, Δ). Its correlation functions are to agree at short distances with asymptotic freedom and perturbative renormalisation.

The Euclidean functional S_YM = (1/4g²)∫𝓕^a_μν𝓕^aμν d⁴x is the integral of a local density, and its stationary points are the connections that satisfy the field equation at every point; its least configurations in each topological sector, the self-dual ones with action 8π²|k|/g², are global objects, labelled by an integer no local variation changes. What organises the theory about the empty configuration is not the functional; it is the premises. The paper names twelve, the core clause that replaces each, and what FDT obtains in its place.

A carrier’s phase is a rotation. Core I-⁠4 reads ψ = Ae^iφ as the projection of a helix onto the plane normal to its axis; in the Clifford algebra of that plane, i is the plane’s unit bivector and e^iφ its rotor. A bivector e_ij = e_ie_j squares to −1, the rotor R = exp(−φe_ij/2) turns the (i, j) plane by φ, and bivectors of planes that share an axis do not commute:

A Yang–Mills connection is a bivector-valued one-form: it states how the carrier’s frame turns along a displacement. Under a change of frame R

𝓕 is the holonomy of the frame per unit area: transport around a small loop returns the frame turned by 𝓕 over the enclosed area, and the term 𝒜∧𝒜 is the composition law of rotations in different planes. One carrier has two presentations here (I-⁠2). Free, its frame turns in one transverse plane, the rotations commute and the field equation is linear: the photon. Standing inside a three-strand closure, its frame turns under the eight bivector generators of the closure’s seat surface, the realisation of su(3) built below; those rotations do not commute: the gluon. The self-coupling is that non-commutation, and nothing is exchanged.

The omnium settles matter, energy and spacetime by one global optimisation, solved at every step from the state the system is in. Events do not commute, a process run backward is never the mirror of the forward one, and responses are system-wide and many to many (0.1a). The core names no objective for the optimisation; the paper reads it through one closure functional of a domain, as its own reading, with every exact book held, and that functional, its per-event response operators and its transition kernel are to be derived (D-⁠01).

The pair law is a face of the settlement. F = F_maxα₁α₂ is exact on two nonoverlapping spherical supports at monopole order, where it returns G𝓜₁𝓜₂/d² (I-⁠1). A resolved configuration supplies its whole matrix of exterior inputs to one response problem at once, and reading the pair outputs as independent and additive is the weak-field, slow-motion reduction (I-⁠4). In a system of cones the selection rule signs the pairs one settlement joins, never pair by pair (P-⁠4). Transitions are hyperedges: a closed mode is stable exactly when every transition to a final state at or below threshold that keeps every exact book vanishes, and beta, 2 → 3 + 3 + 2, is one many-to-many reclosure (I-⁠5b). The problem organises the dynamics about the empty configuration: a power series in g about 𝒜 = 0, built from local vertices of three and four lines joined by propagators. The first result below shows what that expansion cannot hold; the other two show what no sum of pair terms holds.

The gap has the zero series

For SU(N) with b₀ = 11N/3 and the one-loop running μ dg/dμ = −b₀g³/16π², the scale Λ_YM = μ exp(−8π²/b₀g²) does not run, and in a theory of the kind the Clay statement asks for, whose one scale is Λ_YM, every energy of the spectrum is a pure multiple of it, the gap among them. Every derivative of exp(−8π²/b₀g²) is a polynomial in 1/g times the exponential and vanishes as g → 0⁺ (Proposition 1):

The expansion the problem writes into its statement is a power series in g about the empty configuration, and the gap’s series is identically zero. The Clay problem asks for the one quantity its own organising expansion assigns zero at every order. The self-dual sectors enter as exp(−8π²|k|/g²) and are absent at every order as well, and so is the tension of the confining network below, a pure number times Λ_YM²/ħc: its linear term enters at no order. The expansion’s own three-body terms are of another kind; for three static sources they begin at two loops, as powers of the coupling (Brambilla, Ghiglieri and Vairo). The gap belongs to the whole solution.

The Borromean closure is a three-body invariant

Three closed loops can be inseparable with every pairwise linking number zero. The Borromean rings, the closure of the three-braid (𝗌₁𝗌₂⁻¹)³, are Brunnian: remove any one and the other two fall apart, and their linking is Milnor’s triple invariant μ̄(123) = ±1, a three-body quantity no pairwise quantity detects. Any two fibres of one Hopf fibration link once, so three fibres of one fibration are pairwise linked and never Borromean (Proposition 2). The neutral two-strand closure’s exact link is the Hopf pair, native to the Clifford torus, whose (1,1) circles are Hopf fibres; the charged three-strand closure’s leading branch is Borromean (P14). On that branch the closure that holds three seats together is a three-body invariant, and a decomposition into pairs does not contain it.

The confining network is a Steiner minimum

For three sources the minimal network L_Y = min_x Σ|x − q_i| is three segments meeting at 120° when every angle of the triangle is below 120°, the two shorter sides otherwise. No function of one separation gives it as a sum of three pair terms: the equilateral family forces f(s) = s/√3, and two unit sides enclosing 150° have L_Y = 2 against 𝒫/√3 = 2.270. On every triangle (Proposition 3)

The consensus theory’s lattice computation measures the confining term of three static sources on this minimal length: a linear term with the tension of the quark–antiquark string beside a two-body Coulomb term, and a force no superposition of two-body forces reproduces. The bounds place the two-body form: half the perimeter, a sum of pair terms with half the tension, is the network’s lower bound. At short separation the computed potential follows that bound, below the network, and beyond about 0.8 fm between sources its slope passes to that of the network. FDT reads the network as the density geodesics of the standing carrier in a three-seat closure, the loci of P-⁠2 where the transverse density gradient vanishes. Three lines of one tension balance at the junction only where their pulls cancel, at 120°, which is the condition that the network’s length is stationary: with one tension the network’s length is the Steiner minimum. The tension, its equality across two- and three-seat closures and the passage from the bound to the network near 0.8 fm are to be derived (D-⁠06).

Yang–Mills takes its group as an input, and the Clay statement ranges over all of them. FDT reads the group off two structures: the support S³ ⊂ ℂ² of the atlas, with share |z₁|² = α (0.3a), and the three seats of the three-strand closure, with the seat lattice K_∩ = A₂ (I-⁠5a). The three-sphere is the shape of the atlas, never a container.

The seam is the one minimal torus

Over each share 0 < α < 1 lies one torus 𝕋_α = {|z₁|² = α}, and over the two ends lie the two Hopf core circles: the open interval of I-⁠6 is the interior of the moment segment, and its ends are circles. Every 𝕋_α is flat, and (Proposition 4)

The mean curvature vanishes, and the area is stationary and largest, only at α = ½: the seam of 0.3a, on the A-⁠1 chart at r_f = 2R_s on the exterior branch and R_s/2 on the interior branch, is the one minimal torus of the family, the Clifford torus, and every embedded minimal torus in S³ is congruent to it (Brendle). The seam map 𝔍: α ↦ 1 − α exchanges the two moduli at fixed phases, carries 𝕋_α onto 𝕋_1−α and Hopf fibres to Hopf fibres, and fixes the Clifford torus pointwise: the jacket flip. The coordinate swap (z₁, z₂) ↦ (z₂, z₁) is 𝔍 composed with the exchange of the two phases, and on the seam it fixes only the great circle z₁ = z₂. The branch exchange 𝔦 is a third map: it keeps α and fixes the fold.

Rotors, seats and the group

The six bivectors of Cl(4,0) split into two commuting triples, su(2)₊ and su(2)₋. The complex structure of the support is J = e₁₂ + e₃₄; its commutant is u(1)_J ⊕ su(2)₋, whose first factor turns z₁ and z₂ together along the Hopf fibres and whose second is the rotor algebra of the support, S³ ≅ SU(2). The Weyl element w: (z₁, z₂) ↦ (−z₂, z₁) acts on the share as α ↦ 1 − α, fixes no point of S³, and squares to −1, the nontrivial element of the centre (Proposition 5). The share of a configuration is fixed by its two curvatures on its branch, and no rotor moves a configuration on the atlas: the rotors act on the support pair as the seat–support module carries it, on states and their readings, so no rotor carries a closure of the matter phase into the spacetime phase.

The seat lattice is set on three resolved integral seat sectors with the common mode quotiented, a primitive positive integral pairing and refinement and target independence, the four premises of O-⁠10. Its matrix K_∩ = [[2, −1], [−1, 2]] is the Cartan matrix of su(3), with det K_∩ = 3 and coker K_∩ = ℤ₃, the triality of I-⁠5a; on the three seat phases, ℂ³, the commutant of the complex structure is u(3), and removing the seats’ common mode, the one I-⁠5a quotients, leaves su(3), whose eight generators turn the three seat phases among themselves on the seat surface (Proposition 6).

Put the two together: 𝕎 = ℂ³_seats ⊕ ℂ²_support, with Cl(10,0) and its forty-five bivectors. Under the seat–support premises (P16), the commutant of the two complex structures, less the common mode of the five slots, is su(3) ⊕ su(2) ⊕ u(1)_Y, twelve generators whose centre is hypercharge, the relative phase of the seats and the support. The group of unitary maps of 𝕎 that keep the split and have determinant one is (Theorem 7)

The ℤ₆ is the product of the two centres, triality and the centre of the support’s SU(2), and the gluing condition Y + 𝔱/3 + 𝔡/2 ∈ ℤ, with 𝔱 triality and 𝔡 doubletness, holds on every fermion, conjugate, doublet scalar and gauge field of the record. The algebra is that of the SU(5) and Spin(10) constructions of grand unification (Georgi and Glashow; Georgi; Fritzsch and Minkowski; Baez and Huerta): 𝖦_SM is the subgroup of SU(5) that keeps the split ℂ³ ⊕ ℂ². FDT supplies the two summands, the three seats and the support pair, and the premise that nothing mixes them; grand unification adds the generators that mix them.

One generation from the module

The spinor module of Cl(10,0) is the exterior algebra on the five slots, thirty-two states. On a state with N_s occupied seats and support occupations o₁, o₂ the readings are

The book is integral, the occupations; hypercharge is a reading with the denominators three and two, the seat and support counts, and charge is integral exactly when N_s ≡ 0 mod 3: triality zero, closure to white. The volume element acts as i(−1)^N, so the two halves are the two hands. The even half is one generation of the record together with the singlet ν^c, as left-handed fields, and the odd half its conjugates; every gauge anomaly and the mixed gravitational anomaly cancel on it, and with four doublets the global SU(2) anomaly (Witten) is absent (Theorem 8). The slots are seat and support labels, not strands, and which closures carry which slots is to be derived (D-⁠02).

(i) Frozen geometry. On (ℝ⁴, η), G_μν ≡ 0, so I-⁠3 admits only T_μν ≡ 0. Every state of energy E > 0 has R_s = 2G𝓜/c² > 0 and a geometry that is not η; the one state with no source is the empty state, and the omnium has none. The stage is R_s = 0 at every field radius, the end the exterior branch approaches and never reaches. If general relativity is ignored, the Clay problem statement does not apply to reality.

(ii) No gravitational scale. The floor of 0.1a and its limit as G is deleted are (Proposition 9)

Deleting G deletes the floor, and the same step freezes the geometry: one limit produces the frozen stage and the ultraviolet problem together. With G kept, the ultraviolet question becomes resolution independence over resolutions bounded below by 2ℓ_P, and a bounded support has finitely many carrier modes: with k < π/ℓ_P, Weyl’s count gives 1.0 × 10⁶⁰ for a support of radius 1 fm, 1.0 × 10¹⁰⁵ for 1 m and 1.0 × 10¹⁸³ for 10²⁶ m (Proposition 10).

(iii) Infinite volume. Every admissible region carries energy, and with the floor of the spacetime phase a geometry of infinite volume carries infinite energy, while the Clay vacuum assigns zero to it. An FDT spectral statement is posed on a bounded support within a universal domain.

(iv) An empty vacuum. The omnium has no empty state, and the lowest pattern of a closed pattern’s own region keeps the period h/E_base, which HΩ = 0 sets to infinity. The ground is the background 𝓑, the substrate’s cumulative standing-wave state, read at cosmological scale as the microwave background and at quantum scale as quantum fluctuations. It carries a frame: its dipole, 3362.08 ± 0.99 μK, reads a solar-system velocity of 369.82 ± 0.11 km s⁻¹ relative to it, while the laws keep their local covariance. The gap therefore belongs to a background: the mass of a mode is m = h/(c²T_C,q) in its period, so over the stable closed modes of 𝓑 (Lemma 11)

(v)–(vii) The expansion. The short-distance agreement the statement requires writes the expansion about 𝒜 = 0 into the problem, and that expansion assigns the gap zero at every order. The one-loop coupling g²(μ) = 8π²/(b₀ ln(μ/Λ_YM)) diverges as μ → Λ_YM⁺, a divergence at a finite control value that I-⁠6 excludes: the pole marks where the expansion stops. The measured decrease of the effective coupling at short separation is the repulsive branch of I-⁠1a. The gluon presentation has no free state, and observables are settlements of whole configurations.

(ix) Gauge slices. Slices meet orbits more than once, and for a non-abelian group no continuous global slice exists. FDT’s data are loop data: the holonomies of all loops determine the connection up to a change of frame, and no slice is chosen.

(x)–(xi) The parameters. FDT’s force law has no coupling constant; the maps from a charge, a circulating charge or a colour seat to an I-⁠1 input are the core’s register item O-⁠6, and couplings are readings of them. Under (P13) the scale of the stable-sector gap is the neutrino’s confinement cost in its background. The θ term violates P and CP, and the record bounds the neutron’s electric dipole moment below 1.8 × 10⁻²⁶ e cm, and with it θ̄ = θ + θ_q, the vacuum angle with the phase of the quark-mass determinant: |θ̄| ≲ 10⁻¹⁰. A parity-odd term requires an internal crossing operator (P15); the absence of one on the seat surface is to be derived (D-⁠04), and with it θ = 0 in the seat sector; the neutron moment reads θ̄, so θ = 0 meets it with θ_q = 0, to be derived with D-⁠04.

(xii) Confinement. A configuration closes on its own exactly when ℓ_total ∈ K_∩ℤ², triality zero: the proton (2, −1), the neutron (1, −2), the meson (1, 1) and Δ⁺⁺ (3, 0) close, and a single u, uu and ud do not. Triality zero holds for the octet as for the singlet, and which triality-zero configurations are singlets rests on the A₂ table from cone geometry (O-⁠6). On the four premises of the seat lattice (O-⁠10), closure to white is the physics of confinement; the dynamical question is the cost of a closure, the confinement operator and its spectrum (O-⁠9). Jaffe and Witten place the proof of confinement among the extensions of the problem.

Three statements replace the Clay gap, each a statement about the settlement. Every admitted closed mode has m = ħω/c² > 0, its whole confinement cost. Under the neutrino-minimum premise (P13) the stable closed sector has the gap

and its least mode is the free two-strand neutrino. On the closure sector of a closed pattern’s own region the edge is the step between the two lowest confinement eigenvalues, ε_* = ħ(ω₁ − ω₀), where the carrier stands; for the n = 3 closures, the class that carries colour, its operator and the estimate uniform over the closures are to be derived (D-⁠1 of the companion paper). Membership of the stable sector is the ledger’s verdict, and every candidate decay of the least mode is a many-to-many reclosure, none of them open: four-momentum leaves it no lighter closed final state under (P13), and its fermion class forbids a purely bosonic one (I-⁠5b). Free radiation carries no gap.

Each test names the measurement that would end a result, a premise or a reading of the paper.

1. T-⁠1
   Closure to white. A replicated free fractional charge ends closure to white as the reading of confinement. Searches in bulk matter, at accelerators and in cosmic rays find none.
2. T-⁠2
   The neutrino minimum. A stable closed mode lighter than the lightest neutrino ends the neutrino-minimum premise and the stable-sector gap; a resolved nonzero lightest mass, from beta-endpoint, double-beta or cosmological data, meets it. Oscillations measure nonzero splittings and the absolute scale is bounded above.
3. T-⁠3
   The seat sector’s angle. A strong-sector parity-odd contribution to an electric dipole moment, a nonzero θ̄, ends θ = 0 in the seat sector.
4. T-⁠4
   The split. A measured vector state of quantum numbers (3, 2)_−5/6, the transformation of SU(5) that joins a seat slot to a support slot, ends the seat–support premises and the reading of Theorem 7 as the gauge group; a proton decay alone does not, since its reclosure is the ledger’s verdict.
5. T-⁠5
   The content. A chiral fermion outside the gluing condition ends the module reading of Theorem 8.
6. T-⁠6
   The network. A measured hadron record that requires the confining energy of three sources to depart at large separation from one tension on the minimal network ends the reading of Proposition 3; the consensus theory’s lattice potentials without dynamical quarks are its comparator. With dynamical quarks the potential saturates beyond the screening separation, which the settlement reads as a reclosure into separate closures.

Seven pieces sit outside the derivation and leave every result in it unchanged, each to be derived: the closure functional of a domain over its complete configurations, its response operators, its transition kernel (the core’s O-⁠12) and its short-distance correlators (D-⁠01); which closures carry which slots, the mixing of the fibre phase with the relative phase and the Weinberg angle (D-⁠02); the confinement costs of the 125 GeV state and of the W and Z states (D-⁠03); the absence of an internal crossing operator on the seat surface and the vanishing of the quark-mass phase (D-⁠04); the doublet as a crossing pair and the two hands as crossing chirality (D-⁠05); the tension of the network, its equality across two- and three-seat closures and the passage from the two-body form to the network (D-⁠06); and the count of generations in two books, the lepton rungs at n = 3 and the quark rungs n = 3, 5, 7, 9, never conflated (D-⁠07).

For the same source and data, FDT returns the measured record. Under the seat–support premises, the group that acts faithfully on every field of the record, S(U(3) × U(2)), its hypercharges, its chirality and one anomaly-free generation are returned by the Clifford algebra of the support and the three seats, with the ℤ₆ of the gluing condition. No free fractional charge is observed, and the charge reading of every closed configuration is integral. The consensus theory’s lattice computations are read beside the record as computations, never counted as data: the potential of three static sources carries the linear term on the minimal connecting length with the tension of the quark–antiquark string and a force beyond any two-body superposition, with the two-body form at short separation, a test of the network reading; the glueball spectrum, its lightest scalar near 1.7 GeV, is a test for the confinement operator, never its output. The deep-inelastic scaling record is accepted in full. Classical well-posedness on Minkowski space is recovered as the α_ext → 0⁺ reading of the coupled solution.

The observables FDT adds each rest on an input the paper names: θ = 0 in the seat sector, against the Standard Model, where θ̄ is a free parameter, and the Peccei–Quinn extension, which reaches θ̄ = 0 through an axion; and a strictly positive mass for the lightest neutrino, where the consensus account allows zero. Against grand unification FDT reads the proton with no transformation that joins seats to support, a reading the Standard Model shares; whether a reclosure of the proton is open is the ledger’s verdict. The measured neutron moment and proton lifetime already sit where these statements place them. Where FDT, general relativity and quantum mechanics converge, FDT reproduces their results; where a premise departs from nature, FDT is under no obligation to reproduce it.