What is the yang mills derivation?

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Yang-Mills Mass Gap Derivation

The derivation establishes a positive spectral gap on the φ-lattice from the J-cost functional alone.

Step 1. The φ-inverse identity phi_inv_eq gives φ⁻¹ = φ − 1. Combined with phi_plus_inv, this yields the exact gap value.

Step 2. Jcost_phi_exact proves Jcost(φ) = (√5 − 2)/2. Define massGap ≔ (√5 − 2)/2.

Step 3. massGap_pos and Jcost_phi_pos establish 0 < massGap.

Step 4. Jcost_mono_gt_one shows J is monotone on (1, ∞). Hence spectral_gap_pos_rung and spectral_gap prove: for all n ≠ 0, massGap ≤ Jcost(φⁿ).

Step 5. Gauge configurations are modeled by GaugeBondConfig. gauge_mass_gap shows any non-trivial configuration has total cost > 0. gauge_cost_ge_gap gives the quantitative lower bound massGap.

Step 6. ym_lattice_gap confirms the lattice gap: any excitation r ≠ 1 satisfies Jcost(r) > 0, with vacuum uniqueness at r = 1.

The gap Δ = J(φ) is therefore exact, positive, universal across SU(3)×SU(2)×U(1) on Q₃, and parameter-free.

cited recognition theorems

YangMillsMassGap.Jcost_phi_exact Computes the exact mass gap value J(φ) = (√5 − 2)/2 from the J-cost definition.
YangMillsMassGap.massGap_pos Proves strict positivity of the gap constant.
YangMillsMassGap.spectral_gap Establishes the spectral gap theorem: every non-vacuum φ-ladder rung costs at least Δ.
YangMillsMassGap.gauge_mass_gap Applies the gap to gauge bond configurations, proving positive cost for non-trivial excitations.
YangMillsLatticeFromRS.ym_lattice_gap Confirms the lattice mass gap for any excitation in the five YM sectors.

outside recognition

Aspects Recognition does not yet address:

Continuum limit and non-abelian renormalization required for the full Millennium Prize statement
OS positivity and transfer operator details for continuum stability

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.