tick_phases_equally_spaced
plain-language theorem explainer
The theorem shows that for indices j < k in the 8-tick cycle the ratio of phases equals the complex exponential of the angular step (k-j)π/4. Workers deriving the necessity of complex numbers from Recognition Science's ledger structure cite this when establishing that phases lie on the unit circle. The proof unfolds the tickPhase definition then reduces the difference of exponents via exp_sub and ring arithmetic.
Claim. For $j,k$ in the finite set of eight indices with $j<k$, let the phase function be defined by $t(k)=e^{iπk/4}$. Then $t(k)/t(j)=e^{i(k-j)π/4}$.
background
The module MATH-004 derives the necessity of complex numbers from Recognition Science's 8-tick structure. The 8-tick cycle produces phases at angles 0, π/4, π/2, …, 7π/4. These phases are realized by the function tickPhase(k) = Complex.exp(I * π * k / 4), which places each value on the unit circle. The upstream definition of tickPhase supplies the explicit exponential form used here. The local setting states that rotation cannot be represented in one real dimension, so the ledger phases force the introduction of the imaginary unit.
proof idea
The term proof unfolds tickPhase to obtain two exponentials, rewrites their quotient via the identity exp(a)/exp(b)=exp(a-b), applies congr to reduce to the exponent difference, pushes the casts, and finishes with ring to verify the coefficient (k.val - j.val) * π/4 * I.
why it matters
This result fills the spacing step inside MATH-004, confirming that consecutive phases differ by exactly π/4 and therefore lie equally spaced on the unit circle. It supports the module claim that the eight-tick octave (T7) requires two-dimensional rotation and hence complex numbers. No downstream theorems are recorded, so the declaration closes the local algebraic verification of phase uniformity before the necessity argument proceeds to reals_no_rotation and quantum_requires_complex.
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