fundamentalFrequency
Defines the fundamental frequency of the 8-tick cycle as eight divided by the base tick duration tau0. Researchers deriving phase accumulations or spin-statistics relations in Recognition Science reference this scaling when working with the discrete clock. The definition is a direct arithmetic expression using the imported tau0 constant from the Constants module.
claimThe fundamental frequency associated with the 8-tick cycle is given by $f = 8 / tau_0$, where $tau_0$ is the duration of one fundamental tick in RS-native units.
background
The module sets out the 8-tick structure as the fundamental discrete clock of Recognition Science, with phases at 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4. This structure is stated to underlie spin-statistics (odd/even phase mapping to fermion/boson), CPT symmetry, gauge groups, and quantum phase accumulation. The upstream constant tau0 is defined as the fundamental time unit equal to one tick, with the explicit note that one octave equals 8 ticks as the fundamental evolution period.
proof idea
One-line definition that divides the integer 8 by the imported tau0 constant.
why it matters in Recognition Science
This definition supplies the frequency scale for the eight-tick octave (T7 in the forcing chain) and directly supports the spin-statistics key theorem that links phase k=4 to fermion antisymmetry and phase k=0 to boson symmetry. It provides the base frequency needed for later derivations of CPT symmetry and gauge structure within the 8-tick framework.
scope and limits
- Does not assign a numerical value in physical units.
- Does not prove periodicity or phase relations.
- Does not connect to mass formulas or the Recognition Composition Law.
- Does not derive tau0 itself.
formal statement (Lean)
82noncomputable def fundamentalFrequency : ℝ := 8 / IndisputableMonolith.Constants.tau0
proof body
Definition body.
83
84/-! ## Core 8-Tick Theorems for Physics Derivations -/
85
86/-- **SPIN-STATISTICS KEY THEOREM**:
87 Phase k=4 (half-cycle) gives -1, which is the fermion antisymmetry sign.
88 Phase k=0 (identity) gives 1, which is the boson symmetry sign.
89 This connects 8-tick structure to spin-statistics. -/