Foundation Schrodinger Derivation Omega8 Pow Eq Evolution Factor

The phase factor

The Schrödinger equation is the rule that tells a quantum system how its state changes from one moment to the next. In its familiar continuous form, iħ ∂ψ/∂t = Ĥψ, it describes a smooth flow. The declaration omega8_pow_eq_evolution_factor (omega8 raised to a power equals the evolution factor) states a discrete, exact version of this rule for a system that updates in eight distinct steps.

Classically, the eighth roots of unity are the eight complex numbers that give 1 when raised to the eighth power. They sit evenly spaced on the unit circle, like the hours on a clock face. The declaration equates the k-th power of the primitive eighth root, ω₈ᵏ, with the complex exponential exp(-iEₖτ₀/ħ). Here Eₖ is an energy value, τ₀ is a unit time step, and ħ is the reduced Planck constant. This is the standard quantum-mechanical phase factor for a state of energy Eₖ after time τ₀.

In Recognition Science, the framework models a discrete ledger of recognition events. Its forcing chain derives an eight-tick cycle, and the one-tick evolution operator acts as a cyclic shift on an eight-dimensional complex vector space. The theorem proves that each of the eight basis modes, the discrete Fourier transform modes, evolves by exactly this phase factor under one tick. This is not an approximation; it is an exact equality for the discrete system. The continuous Schrödinger equation then appears as a limit, with a bounded remainder term that vanishes for slow modes.

The declaration is one step in a larger derivation. It relies on the identification of ω₈ᵏ with the phase factor, and it is used to prove the full discrete Schrödinger flow, the Hermitian property of the Hamiltonian, and the nonnegativity of energy. The library's certificate bundles these results, all checked by a machine against a small set of axioms. What the declaration does not claim is that the continuous Schrödinger equation is exact at all scales; it holds in the slow-mode limit. It also does not claim that the recognition cycle is the only way to derive quantum mechanics, only that within this framework the derivation is sound.

/-- The k-th eigenvalue of `cyclic_shift` is exactly the integrated
    Schrödinger evolution factor at one tick:
    `ω₈^k = exp(-i · E_k · τ₀ / ℏ)`.

    This is the algebraic bridge `ω₈ = exp(-iπ/4)` plus the
    definition of `quarterTurnEnergy`. -/
theorem omega8_pow_eq_evolution_factor (k : Fin 8) :
    omega8 ^ k.val =
      Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ)) := by
  -- LHS: omega8^k = exp(k · (-iπ/4))
  have hLHS : omega8 ^ k.val = Complex.exp ((k.val : ℂ) * (-Complex.I * Real.pi / 4)) := by
    simp only [omega8, ← Complex.exp_nat_mul]
  rw [hLHS]
  congr 1
  unfold quarterTurnEnergy
  have hhbar_ne : (hbar : ℂ) ≠ 0 := by
    exact_mod_cast (ne_of_gt hbar_pos)
  have htau_ne : (tau0 : ℂ) ≠ 0 := by
    exact_mod_cast (ne_of_gt tau0_pos)
  push_cast
  field_simp

/-- The one-tick recognition evolution acts on each DFT-8 mode by
    multiplication by `ω₈^k = exp(-iπk/4)`. This is the spectral
    identity `cyclic_shift = ⊕ ω₈^k · I_{mode k}`. -/
theorem eigenmode_evolution_exact (k : Fin 8) :
    cyclic_shift (dft8_mode k) = (omega8 ^ k.val) • dft8_mode k :=
  dft8_shift_eigenvector k

/-- **DISCRETE SCHRÖDINGER (eigenmode form).** For every DFT mode `k`
    and every coefficient `c`, the integrated one-tick evolution is
    exactly `ψ(τ₀) = exp(-i E_k τ₀ / ℏ) · ψ(0)`. -/
theorem discrete_schrodinger_eigenmode (k : Fin 8) (c : ℂ) :
    cyclic_shift (c • dft8_mode k) =
      Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ))
        • (c • dft8_mode k) := by
  rw [eigenmode_evolution_scaled k c]
  rw [omega8_pow_eq_evolution_factor k]

/-- **TAYLOR REMAINDER BOUND.** For each eigenmode `k`, when the
    one-tick phase satisfies `‖ -i E_k τ₀ / ℏ ‖ ≤ 1`, the increment
    `cyclic_shift (c • dft8_mode k) - (c • dft8_mode k) - z • (c • dft8_mode k)`
    differs from the linear-in-τ₀ Schrödinger drift by at most a
    quadratic remainder.

    Specifically, with `z := -i E_k τ₀ / ℏ`, the residual
    `(exp z - 1 - z) · ψ` is bounded by `‖z‖² · ‖ψ‖`. -/
theorem schrodinger_remainder_bound (k : Fin 8) (c : ℂ) (t : Fin 8)
    (hsmall : ‖(-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ))‖ ≤ 1) :
    let z : ℂ := -Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ)
    ‖(Complex.exp z - 1 - z) • (c • dft8_mode k) t‖ ≤
      ‖z‖ ^ 2 * ‖(c • dft8_mode k) t‖ := by
  intro z
  have hbnd := exp_taylor_remainder hsmall
  rw [Pi.smul_apply, smul_eq_mul, norm_mul]
  exact mul_le_mul_of_nonneg_right hbnd (norm_nonneg _)

What this page does not claim

The continuous Schrödinger equation is exact at all scales, not just in the slow-mode limit. The recognition cycle is the only possible origin of quantum mechanics. The declaration itself proves the full Schrödinger equation; it is one step within a larger derivation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/SchrodingerDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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