Foundation Schrodinger Derivation Schrodinger Equation From Rs

The derived equation

The Schrödinger equation, iħ∂ψ/∂t = Ĥψ, is the central equation of quantum mechanics. It describes how the quantum state of a system changes over time. In conventional physics, this equation is a postulate, assumed to be true because it successfully predicts experimental results. In the Recognition Science framework, this equation is not assumed; it is a consequence of a more basic structure.

This structure is a discrete ledger, a record of recognition events that is updated in fixed steps. The framework's forcing chain proves that this ledger must operate on an eight-tick cycle. The declaration schrodinger_equation_from_RS in the machine-checked library of formal theorems shows that the time evolution of a state on this eight-step cycle is exactly described by the Schrödinger equation.

The derivation works in five concrete steps. First, the eight-tick cycle defines a space of signals, each a function from eight positions to complex numbers. Second, a single step of the cycle is a cyclic shift, moving each value to the next position. Third, this shift has a natural set of eight basis modes, each rotating by a fixed phase factor. Fourth, this phase factor is identified with the standard quantum evolution factor, exp(-iEτ₀/ħ), which defines the energy E of each mode. Fifth, by linear superposition, the evolution of any signal is the sum of the evolutions of its modes, which is the Schrödinger equation in its integrated form.

The declaration proves several specific properties. The evolution is unitary, meaning it preserves the total probability. The energy values are real numbers, meaning the Hamiltonian is Hermitian. The energy is non-negative. The evolution obeys the superposition principle. These are all proven as theorems, not assumed.

The continuum form, iħ∂ψ/∂t = Ĥψ, is recovered by taking the slow-mode limit, where the time step τ₀ is small. The proof provides a bound on the error of this approximation, which vanishes in this limit. The discrete form, however, is exact.

This result is a significant step in the Recognition Science program. It shows that a fundamental equation of physics, normally taken as an axiom, can be derived from a more primitive notion of discrete recognition. It does not, however, claim to derive the full content of quantum mechanics, such as the specific form of the Hamiltonian for a particular physical system. It derives the form of the equation, not its specific instances.

/-- **SCHRÖDINGER EQUATION FROM RECOGNITION SCIENCE: ONE-STATEMENT THEOREM.**

  For every DFT-8 eigenmode `k` and every complex amplitude `c`, the
  one-tick recognition evolution is exactly the integrated Schrödinger
  flow at energy `E_k = ℏ · πk / (4τ₀)`:

  `cyclic_shift (c · dft8_mode k) = exp(-i E_k τ₀ / ℏ) · (c · dft8_mode k)`.

  Linearity (`schrodinger_linear`) extends this to every state in
  `Signal8`. Energy `E_k` is real (Hermitian Ĥ_RS) and nonnegative.

  In RS-native units (`ℏ = φ⁻⁵`, `τ₀ = 1`) the eigenvalues are
  `E_k = φ⁻⁵ · πk / 4`. -/
theorem schrodinger_equation_from_RS :
    -- (Forward time evolution)
    (∀ (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)) ∧
    -- (Hermitian generator)
    (∀ k : Fin 8, (quarterTurnEnergy k : ℂ).im = 0) ∧
    -- (Energy ≥ 0)
    (∀ k : Fin 8, 0 ≤ quarterTurnEnergy k) ∧
    -- (Superposition principle)
    (∀ (ψ φ : Signal8) (a b : ℂ),
        cyclic_shift (a • ψ + b • φ) =
          a • cyclic_shift ψ + b • cyclic_shift φ) ∧
    -- (Unitarity per mode)
    (∀ (k : Fin 8) (c : ℂ) (t : Fin 8),
        ‖cyclic_shift (c • dft8_mode k) t‖ = ‖(c • dft8_mode k) t‖) :=
  ⟨discrete_schrodinger_eigenmode, quarterTurnEnergy_real,
    quarterTurnEnergy_nonneg, schrodinger_linear,
    eigenmode_norm_preserved⟩

/-- The Hamiltonian eigenvalues are real: `Ĥ_RS` is Hermitian. -/
theorem quarterTurnEnergy_real (k : Fin 8) :
    (quarterTurnEnergy k : ℂ).im = 0 := by
  simp

/-- Energy nonnegativity: every eigenvalue is `≥ 0`. -/
theorem quarterTurnEnergy_nonneg (k : Fin 8) : 0 ≤ quarterTurnEnergy k := by
  unfold quarterTurnEnergy
  have h1 : 0 ≤ hbar := le_of_lt hbar_pos
  have h2 : 0 ≤ Real.pi * (k.val : ℝ) :=
    mul_nonneg Real.pi_pos.le (Nat.cast_nonneg _)
  have h3 : 0 ≤ hbar * (Real.pi * (k.val : ℝ)) := mul_nonneg h1 h2
  have h4 : 0 < 4 * tau0 := by
    have htau : 0 < tau0 := tau0_pos
    linarith
  exact div_nonneg h3 h4.le

/-- One-tick evolution preserves the norm on each mode (unitarity). -/
theorem eigenmode_norm_preserved (k : Fin 8) (c : ℂ) (t : Fin 8) :
    ‖cyclic_shift (c • dft8_mode k) t‖ = ‖(c • dft8_mode k) t‖ := by
  rw [eigenmode_evolution_scaled k c]
  -- Now: ‖((omega8^k.val) • (c • dft8_mode k)) t‖ = ‖(c • dft8_mode k) t‖
  simp only [Pi.smul_apply, smul_eq_mul, norm_mul]
  have homega : ‖omega8 ^ k.val‖ = 1 := by
    rw [norm_pow, omega8_abs, one_pow]
  rw [homega, one_mul]

What this page does not claim

Not claiming to derive the specific Hamiltonian for any particular physical system. Not claiming to derive the full content of quantum mechanics, only the form of the time-evolution equation. Not claiming that the continuum limit is exact; it is an approximation with a bounded remainder.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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