Yang-Mills Mass Gap

1. Setting

Pure Yang-Mills theory asks whether a non-abelian gauge field can have a vacuum separated from all nontrivial excitations by a strictly positive energy gap. In the Clay formulation this is a statement about the quantum SU(N) theory on four-dimensional spacetime: construct the theory, prove existence, and prove a mass gap. The RS page is not allowed to hide behind the phrase 'mass gap bundle'. It must state the physics: no massless gluonic excitation survives above the vacuum. The Lean module Unification.YangMillsMassGap encodes that statement as a positive recognition cost gap tied to the phi-ladder and the J-cost of a nontrivial gauge-bond configuration.

2. Equations

(E1)

$$ \Delta_{\mathrm{YM}} = \inf(\mathrm{Spec}(H)\setminus{0}) > 0 $$

Mass-gap statement in Hamiltonian language.

(E2)

$$ J(\varphi)=\frac12(\varphi+\varphi^{-1})-1=\varphi-\frac12 $$

RS recognition cost at the phi step.

(E3)

$$ m_{0^{++}}\approx 1.7\ \mathrm{GeV} $$

Empirical/lattice anchor for the lightest pure-glue SU(3) scalar glueball.

3. Prediction or structural target

• Pure SU(3) lightest glueball: predicted 0++ state near 1.7 (GeV); empirical 1.710-1.730 GeV lattice range. Source: Morningstar-Peardon 1999; Chen et al. 2006
• Gauge-sector mass gap: predicted positive (energy); empirical No massless gluonic hadron observed. Source: Lattice QCD and hadron spectroscopy

Lattice QCD supports a positive pure-glue mass gap; the lightest 0++ glueball lies near 1.7 GeV.

4. Formal anchor

The primary anchor is Unification.YangMillsMassGap..

5. What is inside the Lean module

Key theorems:

• phi_inv_eq
• phi_plus_inv
• Jcost_phi_exact
• Jcost_phi_eq_phi_minus_half
• Jcost_phi_eq_massGap
• sqrt5_gt_two
• massGap_pos
• Jcost_phi_pos
• Jcost_mono_gt_one
• phiLadder_pos
• Jcost_phiLadder_zero
• Jcost_phiLadder_symm

Key definitions:

• massGap
• PhiLadder
• GaugeBondConfig
• vacuum
• totalGaugeCost
• isNonTrivial
• IsContractible
• GaugeSectorMassGap

6. Derivation chain

• Unification.YangMillsMassGap - YM mass gap module
• Physics.YangMillsLatticeFromRS - YM lattice from RS

7. Falsifier

A massless color-singlet gluonic excitation in pure-gauge SU(3), or a lattice continuum limit in which the lightest glueball mass tends to zero, refutes this page. In RS terms, that would mean a nontrivial gauge-bond configuration can escape the positive J-cost gap.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit yang-mills-mass-gap, start with the primary Lean anchor Unification.YangMillsMassGap. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.