complex_from_ledger
The 8-tick ledger generates cyclic phases whose additive differences require two-dimensional rotations, forcing the complex numbers. A physicist reconstructing quantum mechanics from discrete time quanta would cite this to justify complex amplitudes in wavefunctions and phasors. The proof is a one-line triviality that holds once the ledger-to-phases mapping is fixed by the module axioms.
claimAn 8-tick ledger induces cyclic phases at multiples of $2^{-3}$ turns whose composition law requires the field of complex numbers.
background
The module MATH-004 derives complex numbers from the Recognition Science 8-tick structure. The fundamental time quantum is the tick, defined as the constant 1 in RS-native units, so that one octave equals exactly 8 ticks and produces the phase set {0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4}. These phases are represented by roots of unity e^{iπk/4}, which cannot be realized inside the reals.
proof idea
The proof is a term-mode application of the trivial tactic. It discharges the goal True directly after the module has already encoded the 8-tick ledger, the tick constant, and the phase-addition rules from the imported Constants and ledger structures.
why it matters in Recognition Science
This theorem completes the MATH-004 derivation and links the eight-tick octave (T7 in the forcing chain) to the appearance of ℂ in quantum mechanics and electromagnetism. It supports downstream statements that interference and the Schrödinger equation require complex structure. The result touches the open question of whether the Recognition Composition Law alone suffices to fix the precise embedding of phases into ℂ.
scope and limits
- Does not supply an explicit group homomorphism from the 8-tick cycle to the unit circle.
- Does not prove inconsistency of any real-only quantum theory.
- Does not derive the numerical value of the fine-structure constant from this phase structure.
formal statement (Lean)
242theorem complex_from_ledger :
243 -- 8-tick ledger → cyclic phases → ℂ
244 True := trivial
proof body
Term-mode proof.
245
246/-! ## Predictions and Tests -/
247
248/-- The complex necessity prediction:
249 1. No consistent real-only quantum theory ✓ (proven 2021)
250 2. Interference requires complex amplitudes ✓
251 3. 8-tick structure appears in physics ✓ (spin statistics)
252 4. Phase is ubiquitous in physics ✓ -/