pith. sign in
theorem

spectral_gap

proved
show as:

Yang-Mills mass gap on the φ-lattice: Δ = J(φ) = (√5 − 2)/2.

module
IndisputableMonolith.Unification.YangMillsMassGap
domain
Unification
line
184 · github
papers citing
2 papers (below)

plain-language theorem explainer

The spectral gap theorem asserts that for every nonzero integer n the recognition cost J(phi^n) is bounded below by the mass gap Delta equal to J(phi). Researchers resolving the Yang-Mills mass gap inside the Recognition Science framework cite this result to obtain a universal, parameter-free lower bound on non-vacuum excitations. The proof reduces the claim to the positive-rung case by a single rewrite to the defining equality for massGap followed by case analysis on the sign of n and symmetry under rung inversion.

Claim. For every integer $n$ with $n ≠ 0$, $Δ ≤ J(φ^n)$, where $Δ = J(φ) = (√5 - 2)/2$ is the mass gap and $J(x) = ½(x + x^{-1}) - 1$ is the recognition cost functional on the golden-ratio ladder.

background

Recognition Science equips the golden-ratio lattice with the cost functional J(x) = ½(x + x^{-1}) - 1, which satisfies the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The phi-ladder is the set of all elements φ^n for n ∈ ℤ; each such element represents a stable recognition configuration. The module derives the Yang-Mills mass gap directly from this cost on the ladder, showing it resolves Millennium problem QG-005 by furnishing an exact positive lower bound Δ = J(φ) that holds uniformly for SU(3), SU(2) and U(1) sectors on the cube lattice Q₃.

proof idea

The tactic proof begins with the rewrite ← Jcost_phi_eq_massGap that replaces the left-hand side by the cost of the unit rung. It then performs rcases le_or_gt 1 n to split into n ≥ 1 and n < 1. The positive branch invokes the monotonicity lemma spectral_gap_pos_rung. The negative branch first obtains n ≤ -1 by omega, rewrites via Jcost_phiLadder_symm to reduce to the positive case, and finishes with spectral_gap_pos_rung.

why it matters

This theorem supplies the central inequality that identifies the mass gap as the minimum excitation energy on the phi-ladder. It is invoked directly by the downstream result mass_gap_is_spatial_minimum in SpacetimeEmergence, which states that the gap sets the minimum spatial excitation energy. Within the framework it completes the derivation of the strict spectral gap from the J-cost functional and the phi-forcing chain (T5 J-uniqueness and T6 phi fixed point), furnishing the exact value Δ = φ - 3/2 that is universal across gauge sectors and falsifiable by observation of any lower-cost phi-ladder excitation.

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papers checked against this theorem (showing 2 of 2)