E_passive
E_passive counts the passive field edges in three-dimensional cube geometry as eleven. Mass modelers cite it when constructing sector-specific B_pow and r0 yardsticks for fermion and boson masses on the phi-ladder. The definition is a direct abbreviation of passive_field_edges applied to dimension D.
claim$E_ {passive} := passive_field_edges(D) = 11$, where passive_field_edges subtracts the single active edge per tick from the total number of edges in the unit cube.
background
The Masses.Anchor module centralizes parameter-free constants derived from cube geometry at D=3. Total edges equal twelve as three times two squared; passive edges are the remainder after removing the one active edge per tick. These passive edges dress the interaction in the Recognition Science mass formulas. Upstream, passive_field_edges is defined as cube_edges d minus active_edges_per_tick, and the key number for D=3 is eleven.
proof idea
This is a one-line abbreviation that applies the passive_field_edges function to the dimension D.
why it matters in Recognition Science
It supplies the factor eleven that enters the geometric seed in alpha derivation, where geometric_seed_factor equals eleven as the product of boundary curvature and passive edges. Downstream it parametrizes B_pow for leptons as negative twenty-two and appears in rung calculations for tau and lepton values. It closes the cube-geometry step in the mass manuscript derivation chain.
scope and limits
- Does not assert agreement with measured particle masses.
- Does not derive the value of spatial dimension D.
- Does not include dynamical evolution of the edges.
- Does not cover higher-dimensional generalizations.
formal statement (Lean)
52abbrev E_passive : ℕ := passive_field_edges D
proof body
Definition body.
53
54/-- Wallpaper groups: 17 -/
used by (40)
-
geometric_seed_factor_eq_11 -
passive_edges_at_D3 -
B_pow -
B_pow_Lepton_eq -
r_lepton_values -
tau -
tau_values -
yardstick -
B_pow_eq_alt -
base_shift_denominator_forced -
base_shift_denominator_forced_no_hk -
base_shift_denominator_scale_forced_no_hc_pos -
base_shift_kn_forced_under_passive_bound -
base_shift_kn_forced_under_passive_bound_no_hk -
base_shift_numerator_offset_forced -
base_shift_weight_split_forced -
ledger_fraction_denominator_forced -
ledger_fraction_denominator_forced_no_hk -
ledger_fraction_denominator_scale_forced -
ledger_fraction_denominator_scale_forced_no_hc_pos -
ledger_fraction_kn_forced_under_passive_bound -
ledger_fraction_kn_forced_under_passive_bound_no_hk -
ledger_fraction_numerator_offset_forced -
ledger_fraction_weight_split_forced -
lepton_integer_slot_iff_bundle_no_hk -
lepton_real_scale_iff_bundle_no_hc_pos -
mass_topology_counts -
o4_channel_closure_certificate -
o4_slot_forcing_certificate -
step_e_mu_channel_split