IndisputableMonolith.Unification.YangMillsMassGap
The YangMillsMassGap module derives the Yang-Mills mass gap from the Recognition Science J-cost evaluated at the golden ratio. Gauge theorists and mass-gap researchers would cite the algebraic link to the phi-ladder. The module assembles a chain of identities beginning from the phi-inverse relation and J-cost monotonicity.
claimThe Yang-Mills mass gap satisfies $\Delta = J(\phi)$ where $J(x) = (x + x^{-1})/2 - 1$ and $\phi$ obeys $\phi^2 = \phi + 1$, with the equality obtained after substituting the inverse identity $\phi^{-1} = \phi - 1$.
background
The module sits inside the unification layer and imports the RS time quantum $\tau_0 = 1$ tick, the J-cost ledger, spatial dimension forcing to $D=3$, the cube-derived gauge group $SU(3)\times SU(2)\times U(1)$, the three fermion generations, and the self-similar forcing of $\phi$. Upstream DimensionForcing states that $D=3$ is forced by the RS framework; PhiForcing states that $\phi$ is forced by self-similarity in a discrete ledger with J-cost.
The central algebraic tool is the phi-inverse identity $\phi^{-1} = \phi - 1$, which follows directly from the quadratic minimal polynomial. Sibling lemmas then equate the J-cost at $\phi$ to the mass-gap expression and establish positivity and monotonicity on the phi-ladder.
proof idea
The module is a collection of algebraic lemmas. The core step applies the phi-inverse identity to simplify $J(\phi)$ and then invokes J-cost monotonicity and positivity to obtain the mass-gap equality. One-line wrappers handle the final substitutions; no external tactics beyond ring normalization and field simplification are required.
why it matters in Recognition Science
The module supplies the concrete mass-gap realization that is imported by SpacetimeEmergence, which forces the full Lorentzian geometry from J-cost. It realizes the T5 J-uniqueness and T6 phi fixed-point steps of the forcing chain inside the gauge sector and closes one link in the mass formula on the phi-ladder.
scope and limits
- Does not derive the mass gap for arbitrary non-abelian gauge groups.
- Does not compute the numerical QCD scale from first principles.
- Does not address dynamical confinement or instanton effects.
- Does not include higher-order quantum corrections to the J-cost.
used by (1)
depends on (7)
declarations in this module (48)
-
theorem
phi_inv_eq -
theorem
phi_plus_inv -
theorem
Jcost_phi_exact -
theorem
Jcost_phi_eq_phi_minus_half -
def
massGap -
theorem
Jcost_phi_eq_massGap -
lemma
sqrt5_gt_two -
theorem
massGap_pos -
theorem
Jcost_phi_pos -
theorem
Jcost_mono_gt_one -
def
PhiLadder -
theorem
phiLadder_pos -
theorem
Jcost_phiLadder_zero -
theorem
Jcost_phiLadder_symm -
theorem
phiLadder_ge_phi -
theorem
phiLadder_gt_one -
theorem
spectral_gap_pos_rung -
theorem
spectral_gap -
theorem
spectral_gap_strict -
structure
GaugeBondConfig -
def
vacuum -
def
totalGaugeCost -
lemma
bond_cost_nonneg -
lemma
bond_le_total -
theorem
vacuum_cost_zero -
def
isNonTrivial -
theorem
gauge_mass_gap -
theorem
gauge_cost_ge_gap -
theorem
vacuum_unique_zero_cost -
def
IsContractible -
theorem
contractible_bond_zero_cost -
theorem
noncontractible_bond_gapped -
theorem
gap_separates_sectors -
theorem
gap_topologically_protected -
theorem
gap_rigidity -
structure
GaugeSectorMassGap -
def
RS_gauge_mass_gaps -
theorem
U1_gapless -
theorem
SU2_SU3_gapped -
theorem
mass_gap_asymmetry -
theorem
massGap_numerical_bound -
def
massGap_falsifier -
theorem
massGap_unfalsified -
structure
YMGapCertificate -
theorem
yang_mills_gap_cert -
theorem
yang_mills_gap_cert_nonempty -
theorem
mass_gap_from_forcing_chain -
theorem
yang_mills_mass_gap_complete