Confinement

6.1 — Why you never see a lone quark

[Structural explanation — qualitative, not quantitative. Confinement is explained from isotropy + substrate stiffness, but the flux tube energy is not derived from {S, B, R, C}. See D1.]

A lone quark would be a break whose manifold orientation is locked to one face at macroscopic scale. An anisotropic eye.

The manifold is isotropic. Three linked faces, no preferred one. At macroscopic scales — many records, many actualisations — the manifold insists on isotropy. The substrate is stiff (λ ≈ 2.15 × 10⁴⁶). It restores isotropy. It does not tolerate macroscopic anisotropy.

A lone colour charge is an orientation that has not been cancelled by its complementary alignments. At short distances — inside the hadron — the anisotropy is local and tolerable. One record can orient one way without violating the manifold’s large-scale isotropy.

At large distances — trying to separate a quark from its partners — the anisotropy would become macroscopic. The substrate resists. The energy stored in the anisotropy grows with distance.

The colour flux tube — the region between separated quarks where the orientation field is stretched, resisting the separation, storing energy proportional to the distance.

Eventually, the energy in the flux tube is enough to create a new quark-antiquark pair. The pair pops into existence (Axiom B: the break has minimum size ε; new breaks can occur). One new quark joins the separated quark. The other joins the remnant. Two colour-neutral hadrons where there was one. The quarks are never free.

Confinement is the manifold’s isotropy enforced by the substrate’s stiffness.

You cannot pull a quark free any more than you can stretch a rubber band forever. The manifold snaps back. Not because a force holds the quark in place — because the shape of the room does not permit macroscopic anisotropy.

[Quantitative derivation of the flux tube energy / string tension from the axioms is an open problem (D1). Here is how to destroy this claim: derive the string tension from {S, B, R, C} and show it contradicts lattice QCD measurements. The argument hands you the weapon.]

6.2 — Asymptotic freedom

[Structural explanation — qualitative. See D3.]

At short distances — inside the hadron, at high energies — the quarks behave as if they are free. The strong force gets weaker at short range. Asymptotic freedom (Gross, Wilczek, Politzer, 1973; Nobel Prize 2004).

At short distances, the orientation is local. One record, one alignment. The anisotropy is confined to a single Planck cell or a few cells. The substrate does not need to enforce isotropy at this scale — the anisotropy is already microscopic. The restoring force is negligible. The quarks move freely.

As the distance increases, the anisotropy grows. The substrate notices. The restoring force strengthens. The coupling increases. The quarks are pulled back.

Asymptotic freedom at short range. Confinement at long range. The same mechanism: the substrate’s response to anisotropy in the orientation. Weak when the anisotropy is small. Strong when the anisotropy is large. You are watching one mechanism produce both behaviours — not two forces, one.

[In standard QCD, this behaviour is captured by the negative β-function of non-Abelian gauge theory. D3 is to reproduce the running coupling α_s(Q²) from the stiffness/isotropy mechanism.]

6.3 — Why SU(3) is not broken

SU(2) × U(1) is broken by Axiom B (AP16). The break distinguishes the sectors. SU(2) is broken. U(1) survives.

SU(3) is NOT broken. Why?

You already know the answer. You have been watching it build since §2.

Proposition 3 (SU(3) is exact). Axiom B breaks SU(2) but does not break SU(3). The strong gauge symmetry is exact and the gluons are massless.

Proof. (1) SU(2) is the symmetry of whether the sectors are distinguishable. Axiom B (ε ∈ ℓ with no σ-image in 𝒫) creates a distinction between the two sectors. Breaks SU(2) (AP16). (2) SU(3) is the symmetry of how the manifold orients around the break (§4). It acts on the orientation state |ε⟩ ∈ ℂ³ — the arrangement of three faces around the break event. (3) Axiom B asserts the existence of ε (the break happens) but contains no clause selecting a preferred orientation. B says: ε ∈ ℓ, ν(ε) = 1, no σ-image. None of these conditions distinguish one face from another. (4) The non-derivability of {C, S, B} (AP20 §4.2, fenced KS-P.2) guarantees that no axiom can induce a preference among the faces. If one face could be preferred, it would imply a derivability relation between axioms. (5) Therefore no term in the action/Lagrangian can break SU(3). The gauge symmetry is exact. The gauge bosons (gluons) acquire no mass.

Two words do different work. SU(2) is the symmetry of whether the break exists. Axiom B breaks it. SU(3) is the symmetry of how the manifold arranges around the break. Axiom B does not break it. The break says “I exist.” It does not say “I point this way.”

SU(3) is exact. Unbroken. The gluons are massless.