wBosonCount
Recognition Science fixes the W boson multiplicity at three by subtracting one from the square of the SU(2) rank. Gauge theorists checking the pre-symmetry-breaking carrier inventory would cite this when summing to the total of twelve bosons. The definition evaluates by direct substitution of the constant rank value two.
claimLet $r$ be the rank of the SU(2) factor. The W boson count is $r^2 - 1$.
background
The module derives the Standard Model gauge group SU(3)×SU(2)×U(1) from Recognition Science rank decomposition, with SU(3) rank equal to spatial dimension D=3, SU(2) rank equal to D-1, and U(1) rank 1. The upstream rankSU2 definition supplies the fixed value 2 for all multiplicity calculations in this file. The module documentation states that five boson types match configDim D=5 while the explicit count before EWSB reaches twelve.
proof idea
One-line definition that substitutes the upstream rankSU2 value of 2 into the arithmetic expression rank squared minus one.
why it matters in Recognition Science
The definition supplies the SU(2) contribution to totalCarriers, which sums gluonCount plus this term plus rankU1 to reach twelve carriers before EWSB. It completes the gauge boson counting step in the A1 SM Depth module, consistent with the forcing chain step that sets D=3 and the eight-tick octave. The downstream theorem w_boson_count confirms the numerical result of 3.
scope and limits
- Does not derive the SU(2) rank from the forcing chain.
- Does not incorporate electroweak symmetry breaking.
- Does not count the Z boson or photon.
- Does not address coupling constants or masses.
Lean usage
def totalCarriers : ℕ := gluonCount + wBosonCount + rankU1formal statement (Lean)
41def wBosonCount : ℕ := rankSU2 ^ 2 - 1