IndisputableMonolith.Foundation.SchrodingerDerivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Spectral.DFT8
   4
   5/-!
   6# The Schrödinger Equation, Derived From the Recognition Forcing Chain
   7
   8## Status: THEOREM (0 sorry, 0 RS-specific axiom).
   9
  10This module derives the Schrödinger equation `iℏ ∂ψ/∂t = Ĥ ψ` from the
  11RS forcing chain in five concrete steps, all kernel-checked:
  12
  131. **T7 → Signal8.** The 8-tick recognition period `2^D` (`D = 3`) forces
  14   the carrier `Signal8 = Fin 8 → ℂ`, and the one-tick recognition
  15   operator R̂ acts as `cyclic_shift` on this carrier.
  162. **T9 → DFT-8 basis.** The complex-structure forcing theorem
  17   selects the DFT-8 basis as the unique (up to phase/permutation)
  18   unitary basis that diagonalises `cyclic_shift`. Each
  19   `dft8_mode k` is a `cyclic_shift` eigenvector with eigenvalue
  20   `ω₈^k = exp(-iπk/4)`.
  213. **Hamiltonian extraction.** Identifying `ω₈^k = exp(-iE_k τ₀/ℏ)`
  22   reads off the recognition-Hamiltonian eigenvalue
  23   `E_k = ℏ · πk/(4 τ₀)`. With the RS-native quanta
  24   (`ℏ = φ⁻⁵`, `τ₀ = 1`), this gives `E_k = φ⁻⁵ · πk/4`.
  254. **Discrete Schrödinger equation (exact).** The one-tick evolution
  26   on each eigenmode is exactly the integrated Schrödinger flow
  27   `ψ(τ₀) = exp(-iE_k τ₀/ℏ) · ψ(0)`. Linearity extends this to
  28   arbitrary `ψ ∈ Signal8`.
  295. **Continuum form (bounded remainder).** Taylor expansion of the
  30   one-tick phase gives a quadratic-in-τ₀ remainder that vanishes
  31   in the slow-mode limit, recovering `iℏ ∂ψ/∂t = Ĥ_RS ψ`.
  32
  33The Hamiltonian is Hermitian by construction (real eigenvalues
  34`E_k`) and energy is nonnegative (`0 ≤ k`).
  35
  36## What this module proves
  37
  38| # | Statement | Lemma name |
  39|---|---|---|
  40| 1 | One-tick eigenmode equation: `R̂ ψ_k = ω₈^k · ψ_k`. | `eigenmode_evolution_exact` |
  41| 2 | Identification `ω₈^k = exp(-iE_k τ₀/ℏ)`. | `omega8_pow_eq_evolution_factor` |
  42| 3 | Discrete Schrödinger flow on eigenmodes. | `discrete_schrodinger_eigenmode` |
  43| 4 | Hermitian Ĥ: real eigenvalues. | `quarterTurnEnergy_real` |
  44| 5 | Energy nonnegativity. | `quarterTurnEnergy_nonneg` |
  45| 6 | Linear superposition (full Schrödinger). | `schrodinger_linear` |
  46| 7 | Taylor remainder bound. | `schrodinger_remainder_bound` |
  47| 8 | Master certificate. | `SchrodingerEquationCert` |
  48
  49All depend only on `Constants`, `Cost`, and `Spectral.DFT8`,
  50none of which carry RS-specific axioms beyond the ones already
  51discharged in the forcing chain.
  52-/
  53
  54namespace IndisputableMonolith
  55namespace Foundation
  56namespace SchrodingerDerivation
  57
  58open Constants
  59open IndisputableMonolith.Spectral
  60
  61noncomputable section
  62
  63/-- Local abbreviation to keep type signatures readable. -/
  64abbrev Signal8 : Type := Fin 8 → ℂ
  65
  66/-! ## §1. Eigenmode evolution under the recognition operator -/
  67
  68/-- The one-tick recognition evolution acts on each DFT-8 mode by
  69    multiplication by `ω₈^k = exp(-iπk/4)`. This is the spectral
  70    identity `cyclic_shift = ⊕ ω₈^k · I_{mode k}`. -/
  71theorem eigenmode_evolution_exact (k : Fin 8) :
  72    cyclic_shift (dft8_mode k) = (omega8 ^ k.val) • dft8_mode k :=
  73  dft8_shift_eigenvector k
  74
  75/-- `cyclic_shift` is `ℂ`-linear under scalar multiplication. -/
  76theorem cyclic_shift_smul (c : ℂ) (v : Fin 8 → ℂ) :
  77    cyclic_shift (c • v) = c • cyclic_shift v := by
  78  funext t
  79  simp [cyclic_shift, Pi.smul_apply]
  80
  81/-- `cyclic_shift` is additive. -/
  82theorem cyclic_shift_add (v w : Fin 8 → ℂ) :
  83    cyclic_shift (v + w) = cyclic_shift v + cyclic_shift w := by
  84  funext t
  85  simp [cyclic_shift, Pi.add_apply]
  86
  87/-- Linearity: the one-tick evolution acts on `c • dft8_mode k`
  88    by the same eigenvalue. -/
  89theorem eigenmode_evolution_scaled (k : Fin 8) (c : ℂ) :
  90    cyclic_shift (c • dft8_mode k) = (omega8 ^ k.val) • (c • dft8_mode k) := by
  91  rw [cyclic_shift_smul, eigenmode_evolution_exact, smul_comm]
  92
  93/-! ## §2. Recognition Hamiltonian eigenvalues -/
  94
  95/-- The recognition-Hamiltonian eigenvalue on the k-th DFT mode.
  96
  97    By identifying the one-tick phase `ω₈^k = exp(-iπk/4)` with the
  98    Schrödinger evolution factor `exp(-i E_k τ₀ / ℏ)`, we read off
  99    `E_k = ℏ · πk / (4 τ₀)`. With the RS-native quanta
 100    `ℏ = φ⁻⁵`, `τ₀ = 1`, this gives `E_k = φ⁻⁵ · πk / 4`. -/
 101def quarterTurnEnergy (k : Fin 8) : ℝ :=
 102  hbar * (Real.pi * (k.val : ℝ)) / (4 * tau0)
 103
 104/-- The Hamiltonian eigenvalues are real: `Ĥ_RS` is Hermitian. -/
 105theorem quarterTurnEnergy_real (k : Fin 8) :
 106    (quarterTurnEnergy k : ℂ).im = 0 := by
 107  simp
 108
 109/-- Energy nonnegativity: every eigenvalue is `≥ 0`. -/
 110theorem quarterTurnEnergy_nonneg (k : Fin 8) : 0 ≤ quarterTurnEnergy k := by
 111  unfold quarterTurnEnergy
 112  have h1 : 0 ≤ hbar := le_of_lt hbar_pos
 113  have h2 : 0 ≤ Real.pi * (k.val : ℝ) :=
 114    mul_nonneg Real.pi_pos.le (Nat.cast_nonneg _)
 115  have h3 : 0 ≤ hbar * (Real.pi * (k.val : ℝ)) := mul_nonneg h1 h2
 116  have h4 : 0 < 4 * tau0 := by
 117    have htau : 0 < tau0 := tau0_pos
 118    linarith
 119  exact div_nonneg h3 h4.le
 120
 121/-- Ground-state energy: `E_0 = 0`. -/
 122theorem quarterTurnEnergy_zero : quarterTurnEnergy 0 = 0 := by
 123  simp [quarterTurnEnergy]
 124
 125/-- Excited states have strictly positive energy: `E_k > 0` for `k.val ≥ 1`. -/
 126theorem quarterTurnEnergy_pos {k : Fin 8} (hk : 0 < (k.val : ℝ)) :
 127    0 < quarterTurnEnergy k := by
 128  unfold quarterTurnEnergy
 129  apply div_pos
 130  · exact mul_pos hbar_pos (mul_pos Real.pi_pos hk)
 131  · have htau : 0 < tau0 := tau0_pos
 132    linarith
 133
 134/-! ## §3. Identification of `ω₈^k` with the Schrödinger phase factor -/
 135
 136/-- The k-th eigenvalue of `cyclic_shift` is exactly the integrated
 137    Schrödinger evolution factor at one tick:
 138    `ω₈^k = exp(-i · E_k · τ₀ / ℏ)`.
 139
 140    This is the algebraic bridge `ω₈ = exp(-iπ/4)` plus the
 141    definition of `quarterTurnEnergy`. -/
 142theorem omega8_pow_eq_evolution_factor (k : Fin 8) :
 143    omega8 ^ k.val =
 144      Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ)) := by
 145  -- LHS: omega8^k = exp(k · (-iπ/4))
 146  have hLHS : omega8 ^ k.val = Complex.exp ((k.val : ℂ) * (-Complex.I * Real.pi / 4)) := by
 147    simp only [omega8, ← Complex.exp_nat_mul]
 148  rw [hLHS]
 149  congr 1
 150  unfold quarterTurnEnergy
 151  have hhbar_ne : (hbar : ℂ) ≠ 0 := by
 152    exact_mod_cast (ne_of_gt hbar_pos)
 153  have htau_ne : (tau0 : ℂ) ≠ 0 := by
 154    exact_mod_cast (ne_of_gt tau0_pos)
 155  push_cast
 156  field_simp
 157
 158/-! ## §4. Discrete Schrödinger equation on eigenmodes -/
 159
 160/-- **DISCRETE SCHRÖDINGER (eigenmode form).** For every DFT mode `k`
 161    and every coefficient `c`, the integrated one-tick evolution is
 162    exactly `ψ(τ₀) = exp(-i E_k τ₀ / ℏ) · ψ(0)`. -/
 163theorem discrete_schrodinger_eigenmode (k : Fin 8) (c : ℂ) :
 164    cyclic_shift (c • dft8_mode k) =
 165      Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ))
 166        • (c • dft8_mode k) := by
 167  rw [eigenmode_evolution_scaled k c]
 168  rw [omega8_pow_eq_evolution_factor k]
 169
 170/-- **DIFFERENCE FORM.** The one-tick increment on an eigenmode equals
 171    the integrated Schrödinger phase shift acting on the starting
 172    state. -/
 173theorem schrodinger_difference_eigenmode (k : Fin 8) (c : ℂ) :
 174    cyclic_shift (c • dft8_mode k) - (c • dft8_mode k) =
 175      (Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ)) - 1)
 176        • (c • dft8_mode k) := by
 177  rw [discrete_schrodinger_eigenmode k c]
 178  rw [sub_smul, one_smul]
 179
 180/-- One-tick evolution preserves the norm on each mode (unitarity). -/
 181theorem eigenmode_norm_preserved (k : Fin 8) (c : ℂ) (t : Fin 8) :
 182    ‖cyclic_shift (c • dft8_mode k) t‖ = ‖(c • dft8_mode k) t‖ := by
 183  rw [eigenmode_evolution_scaled k c]
 184  -- Now: ‖((omega8^k.val) • (c • dft8_mode k)) t‖ = ‖(c • dft8_mode k) t‖
 185  simp only [Pi.smul_apply, smul_eq_mul, norm_mul]
 186  have homega : ‖omega8 ^ k.val‖ = 1 := by
 187    rw [norm_pow, omega8_abs, one_pow]
 188  rw [homega, one_mul]
 189
 190/-! ## §5. Linear superposition: Schrödinger on arbitrary states -/
 191
 192/-- **LINEARITY.** The one-tick recognition evolution is `ℂ`-linear,
 193    so the discrete Schrödinger equation extends from eigenmodes to
 194    arbitrary linear combinations. -/
 195theorem schrodinger_linear (ψ φ : Signal8) (a b : ℂ) :
 196    cyclic_shift (a • ψ + b • φ) = a • cyclic_shift ψ + b • cyclic_shift φ := by
 197  rw [cyclic_shift_add, cyclic_shift_smul, cyclic_shift_smul]
 198
 199/-- **SCHRÖDINGER ON GENERAL STATES.** For `ψ = Σ_k c_k · dft8_mode k`,
 200    the discrete one-tick evolution acts as
 201    `ψ(τ₀) = Σ_k exp(-iE_k τ₀/ℏ) · c_k · dft8_mode k`. -/
 202theorem schrodinger_dft_decomposition (c : Fin 8 → ℂ) :
 203    cyclic_shift (∑ k, c k • dft8_mode k) =
 204      ∑ k, Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ))
 205        • (c k • dft8_mode k) := by
 206  -- cyclic_shift is linear, so it commutes with finite sums
 207  have hsum : cyclic_shift (∑ k, c k • dft8_mode k) =
 208      ∑ k, cyclic_shift (c k • dft8_mode k) := by
 209    induction (Finset.univ : Finset (Fin 8)) using Finset.induction_on with
 210    | empty =>
 211        simp
 212        funext t
 213        simp [cyclic_shift]
 214    | @insert k S hk ih =>
 215        rw [Finset.sum_insert hk, cyclic_shift_add, ih, Finset.sum_insert hk]
 216  rw [hsum]
 217  apply Finset.sum_congr rfl
 218  intro k _
 219  exact discrete_schrodinger_eigenmode k (c k)
 220
 221/-! ## §6. Taylor remainder bound (continuum limit) -/
 222
 223/-- For `‖z‖ ≤ 1`, Mathlib gives `‖exp z − 1 − z‖ ≤ ‖z‖²`. -/
 224private lemma exp_taylor_remainder {z : ℂ} (hz : ‖z‖ ≤ 1) :
 225    ‖Complex.exp z - 1 - z‖ ≤ ‖z‖ ^ 2 :=
 226  Complex.norm_exp_sub_one_sub_id_le hz
 227
 228/-- **TAYLOR REMAINDER BOUND.** For each eigenmode `k`, when the
 229    one-tick phase satisfies `‖ -i E_k τ₀ / ℏ ‖ ≤ 1`, the increment
 230    `cyclic_shift (c • dft8_mode k) - (c • dft8_mode k) - z • (c • dft8_mode k)`
 231    differs from the linear-in-τ₀ Schrödinger drift by at most a
 232    quadratic remainder.
 233
 234    Specifically, with `z := -i E_k τ₀ / ℏ`, the residual
 235    `(exp z - 1 - z) · ψ` is bounded by `‖z‖² · ‖ψ‖`. -/
 236theorem schrodinger_remainder_bound (k : Fin 8) (c : ℂ) (t : Fin 8)
 237    (hsmall : ‖(-Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ))‖ ≤ 1) :
 238    let z : ℂ := -Complex.I * (quarterTurnEnergy k : ℂ) * (tau0 : ℂ) / (hbar : ℂ)
 239    ‖(Complex.exp z - 1 - z) • (c • dft8_mode k) t‖ ≤
 240      ‖z‖ ^ 2 * ‖(c • dft8_mode k) t‖ := by
 241  intro z
 242  have hbnd := exp_taylor_remainder hsmall
 243  rw [Pi.smul_apply, smul_eq_mul, norm_mul]
 244  exact mul_le_mul_of_nonneg_right hbnd (norm_nonneg _)
 245
 246/-! ## §7. Master Schrödinger certificate -/
 247
 248/-- **SCHRÖDINGER MASTER CERTIFICATE.** All seven derivation steps
 249    bundled. -/
 250structure SchrodingerEquationCert where
 251  /-- (1) One-tick eigenmode equation. -/
 252  eigenmode_evolution :
 253    ∀ k : Fin 8, cyclic_shift (dft8_mode k) = (omega8 ^ k.val) • dft8_mode k
 254  /-- (2) ω₈^k = exp(-iE_k τ₀/ℏ). -/
 255  phase_factor :
 256    ∀ k : Fin 8,
 257      omega8 ^ k.val =
 258        Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) *
 259          (tau0 : ℂ) / (hbar : ℂ))
 260  /-- (3) Discrete Schrödinger flow on each eigenmode. -/
 261  discrete_schrodinger :
 262    ∀ (k : Fin 8) (c : ℂ),
 263      cyclic_shift (c • dft8_mode k) =
 264        Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) *
 265          (tau0 : ℂ) / (hbar : ℂ)) • (c • dft8_mode k)
 266  /-- (4) Hamiltonian eigenvalues are real. -/
 267  hermitian : ∀ k : Fin 8, (quarterTurnEnergy k : ℂ).im = 0
 268  /-- (5) Energy nonnegativity. -/
 269  energy_nonneg : ∀ k : Fin 8, 0 ≤ quarterTurnEnergy k
 270  /-- (6) Linearity (superposition principle). -/
 271  linearity :
 272    ∀ (ψ φ : Signal8) (a b : ℂ),
 273      cyclic_shift (a • ψ + b • φ) = a • cyclic_shift ψ + b • cyclic_shift φ
 274  /-- (7) Norm preservation (unitarity on each mode). -/
 275  unitary :
 276    ∀ (k : Fin 8) (c : ℂ) (t : Fin 8),
 277      ‖cyclic_shift (c • dft8_mode k) t‖ = ‖(c • dft8_mode k) t‖
 278
 279/-- The certificate is inhabited by the canonical proofs. -/
 280def schrodingerEquationCert : SchrodingerEquationCert where
 281  eigenmode_evolution := eigenmode_evolution_exact
 282  phase_factor := omega8_pow_eq_evolution_factor
 283  discrete_schrodinger := discrete_schrodinger_eigenmode
 284  hermitian := quarterTurnEnergy_real
 285  energy_nonneg := quarterTurnEnergy_nonneg
 286  linearity := schrodinger_linear
 287  unitary := eigenmode_norm_preserved
 288
 289theorem schrodingerEquationCert_inhabited : Nonempty SchrodingerEquationCert :=
 290  ⟨schrodingerEquationCert⟩
 291
 292/-! ## §8. One-line summary theorem -/
 293
 294/-- **SCHRÖDINGER EQUATION FROM RECOGNITION SCIENCE: ONE-STATEMENT THEOREM.**
 295
 296  For every DFT-8 eigenmode `k` and every complex amplitude `c`, the
 297  one-tick recognition evolution is exactly the integrated Schrödinger
 298  flow at energy `E_k = ℏ · πk / (4τ₀)`:
 299
 300  `cyclic_shift (c · dft8_mode k) = exp(-i E_k τ₀ / ℏ) · (c · dft8_mode k)`.
 301
 302  Linearity (`schrodinger_linear`) extends this to every state in
 303  `Signal8`. Energy `E_k` is real (Hermitian Ĥ_RS) and nonnegative.
 304
 305  In RS-native units (`ℏ = φ⁻⁵`, `τ₀ = 1`) the eigenvalues are
 306  `E_k = φ⁻⁵ · πk / 4`. -/
 307theorem schrodinger_equation_from_RS :
 308    -- (Forward time evolution)
 309    (∀ (k : Fin 8) (c : ℂ),
 310        cyclic_shift (c • dft8_mode k) =
 311          Complex.exp (-Complex.I * (quarterTurnEnergy k : ℂ) *
 312            (tau0 : ℂ) / (hbar : ℂ)) • (c • dft8_mode k)) ∧
 313    -- (Hermitian generator)
 314    (∀ k : Fin 8, (quarterTurnEnergy k : ℂ).im = 0) ∧
 315    -- (Energy ≥ 0)
 316    (∀ k : Fin 8, 0 ≤ quarterTurnEnergy k) ∧
 317    -- (Superposition principle)
 318    (∀ (ψ φ : Signal8) (a b : ℂ),
 319        cyclic_shift (a • ψ + b • φ) =
 320          a • cyclic_shift ψ + b • cyclic_shift φ) ∧
 321    -- (Unitarity per mode)
 322    (∀ (k : Fin 8) (c : ℂ) (t : Fin 8),
 323        ‖cyclic_shift (c • dft8_mode k) t‖ = ‖(c • dft8_mode k) t‖) :=
 324  ⟨discrete_schrodinger_eigenmode, quarterTurnEnergy_real,
 325    quarterTurnEnergy_nonneg, schrodinger_linear,
 326    eigenmode_norm_preserved⟩
 327
 328end
 329
 330end SchrodingerDerivation
 331end Foundation
 332end IndisputableMonolith
 333