IndisputableMonolith.Unification.YangMillsMassGap
The YangMillsMassGap module derives a positive lower bound on the Yang-Mills mass gap from the J-cost evaluated on the golden ratio phi within the Recognition Science framework. Physicists studying the Clay Millennium problem would cite it for the RS-native derivation of massGap > 0. The argument proceeds by algebraic identities on phi followed by positivity and monotonicity lemmas on the cost function.
claimThe mass gap satisfies $massGap = Jcost_phi > 0$, where $J(x) = (x + x^{-1})/2 - 1$ and $phi$ obeys the identity $phi^{-1} = phi - 1$. The phi-ladder $PhiLadder$ supplies the discrete scale structure on which the gap is realized.
background
The module sits in the unification layer and imports the J-cost definition from Cost, the self-similar forcing of phi from PhiForcing, and the gauge group structure from GaugeFromCube. It introduces the phi-ladder PhiLadder as the discrete scale hierarchy and defines massGap directly from the J-cost at the phi rung. Upstream PhiForcing states that phi is forced as the self-similar fixed point of a discrete ledger equipped with J-cost. The supplied phi-inverse identity follows at once from the quadratic equation $phi^2 = phi + 1$.
proof idea
The module first records the elementary identities phi_inv_eq and phi_plus_inv. It then evaluates the exact J-cost expression Jcost_phi_exact, reduces it via Jcost_phi_eq_phi_minus_half, and equates the result to massGap by Jcost_phi_eq_massGap. Positivity is obtained from sqrt5_gt_two together with the dedicated lemmas massGap_pos and Jcost_phi_pos; monotonicity Jcost_mono_gt_one ensures the gap is the minimal positive value on the ladder. PhiLadder and phiLadder_pos close the construction.
why it matters in Recognition Science
This module supplies the mass-gap ingredient required by the downstream SpacetimeEmergence module, which forces the full Lorentzian structure from J-cost. It fills the Yang-Mills mass-gap slot in the unification chain and links directly to the phi-ladder mass formula of the Recognition Science framework. It touches the open question of extracting a numerical value for the gap in RS-native units.
scope and limits
- Does not prove existence of Yang-Mills fields or the full quantum theory.
- Does not compute a numerical value for the gap beyond positivity.
- Does not address confinement dynamics beyond the mass-gap bound.
- Does not connect the RS gap to experimental glueball spectra.
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