Explain the Lean theorem `spectral_gap` in module `IndisputableMonolith.Unification.YangMillsMassGap`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Plain English statement

The theorem spectral_gap states that for any integer n ≠ 0, the RS mass gap constant satisfies massGap ≤ Jcost(PhiLadder n). In plain terms: every non-vacuum rung on the golden-ratio lattice φ^n carries J-cost at least Δ = J(φ) = (√5 − 2)/2 > 0.

Relevance in Recognition Science

It supplies the exact positive lower bound on excitation cost for the φ-lattice substrate. This is the structural core of the RS resolution of the Yang-Mills mass gap (QG-005): non-trivial gauge bonds on Q₃ cannot approach zero cost, with the bound derived solely from the J-cost functional and φ-forcing.

How to read the formal statement

 theorem spectral_gap (n : ℤ) (hn : n ≠ 0) :
     massGap ≤ Jcost (PhiLadder n)

Parameters: integer rung n and proof hn that n ≠ 0. Conclusion: inequality between the pre-defined massGap and the J-cost of the corresponding ladder element. The proof rewrites via Jcost_phi_eq_massGap, then cases on sign of n, invoking symmetry for negatives.

Visible dependencies or certificates

• Relies on spectral_gap_pos_rung for n ≥ 1 and Jcost_phiLadder_symm for n ≤ −1.
• Forms part of the gauge_mass_gap and complete Yang-Mills mass gap certificate in the same module (zero sorry).

What this declaration does not prove

It establishes the gap only on the discrete φ-lattice; it does not derive the gauge groups, perform the continuum limit, or address non-abelian renormalization.