module module high

`IndisputableMonolith.Mathematics.ComplexNumbers`

The module supplies the complex number framework for representing the eight phases of the recognition tick cycle. Researchers deriving the discrete time structure from the unified forcing chain would cite these constructions. It proceeds through definitions of phases as roots of unity, lemmas showing real numbers are insufficient, and extensions to phasors and Fourier transforms.

claimThe eight phases of the tick cycle are the eighth roots of unity in the complex plane, satisfying $z^8 = 1$ with the fundamental tick quantum satisfying $τ_0 = 1$.

background

The module imports the fundamental RS time quantum $τ_0 = 1$ tick from Constants. It introduces tickPhase as the complex representation of phases in the cycle and related constructions such as roots of unity and phasors. The local theoretical setting is the mathematics of rotations and oscillations required once the forcing chain reaches the eight-tick octave.

proof idea

This is a definition module containing a sequence of definitions and supporting lemmas. It begins with the phase definition, establishes the roots-of-unity property, proves that real numbers cannot encode the required rotations, and extends the constructions to quantum and Fourier applications.

why it matters in Recognition Science

The module supplies the phase machinery that realizes T7 (eight-tick octave) in the forcing chain. It directly enables downstream constructions such as the Schrödinger equation and Fourier transforms that rely on complex phases within the Recognition Science framework.

scope and limits

Does not derive any step of the forcing chain T0-T8.
Does not compute values of physical constants such as alpha or G.
Does not address spatial dimensions or the mass ladder.
Does not treat Berry creation or Z_cf thresholds.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (21)

def tickPhase L50
theorem tick_phases_roots_of_unity L54
theorem tick_phases_equally_spaced L67
theorem reals_no_rotation L82
theorem complex_rotation L91
theorem phases_require_complex_k1 L104
theorem phases_require_complex_k2 L117
theorem phases_require_complex L128
theorem quantum_requires_complex L164
def schrodingerEquation L171
def phasor L176
theorem fourier_uses_complex L181
theorem complex_inevitable L196
theorem euler_formula L203
theorem quaternions_not_needed L213
theorem split_complex_insufficient L220
theorem complex_is_unique L227
theorem complex_from_ledger L242
def predictions L253
structure ComplexFalsifier L267
def experimentalStatus L274