hypercharge_layer
The hypercharge_layer definition supplies the U(1) component in the three-layer decomposition of the 3-cube automorphism group B3. Gauge theorists deriving the Standard Model group from discrete geometry cite it to complete the (3,2,1) rank triple. It is a direct record construction assigning fundamental representation dimension 1 and discrete order 2.
claimThe hypercharge gauge layer is the record with name ``U(1) hypercharge'', fundamental representation dimension 1, and discrete order 2.
background
The GaugeLayer structure records the name, fundamental representation dimension, and discrete order for each gauge component. In this module the 3-cube Q3 has automorphism group B3 of order 48, which decomposes into axis permutations S3 (color, dimension 3), even sign flips of order 4 (weak, dimension 2), and the remaining parity factor (hypercharge, dimension 1). The module builds on prior derivations of three generations from face-pairs and three colors from the same geometry.
proof idea
This is a definition that directly constructs the GaugeLayer record with the three fields set to the hypercharge values. No lemmas or tactics are invoked; the assignment completes the layer triple used by downstream rank and order calculations.
why it matters in Recognition Science
The definition supplies the missing U(1) entry required by gauge_group_certificate, dimension_sum, and gauge_order_product. Those theorems establish that the ranks (3,2,1) and orders (6,4,2) reproduce the Standard Model group and the order of B3. Within Recognition Science this realizes the D=3 triangular decomposition 3+2+1=6 of face-pair interactions.
scope and limits
- Does not derive the hypercharge coupling strength or particle charges.
- Does not address quantum corrections or renormalization group flow.
- Does not connect the layer to the phi-ladder mass formula.
- Does not prove uniqueness of this assignment among possible discrete groups.
formal statement (Lean)
254def hypercharge_layer : GaugeLayer :=
proof body
Definition body.
255 { name := "U(1) hypercharge"
256 fund_rep_dim := 1
257 discrete_order := 2 }
258
259/-- **THEOREM (Gauge Rank Match)**:
260 The three layers of B₃ have fundamental representation dimensions
261 (3, 2, 1) — matching the Standard Model gauge group SU(3) × SU(2) × U(1). -/